In probability theory, a normal (or Gaussian or Gauss or Laplace-Gauss) distribution is a type of continuous probability distribution for a real-valued random variable.The general form of its probability density function is = ()The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation Normal random variables Now that we have seen the standard normal random variable, we can obtain any normal random variable by shifting and scaling a standard normal random variable. In particular, define X = σZ + μ, where σ > 0. Then EX = σEZ + μ = μ, Var(X) = σ2Var(Z) = σ2 For any normal random variable, if you find the Z-score for a value (i.e standardize the value), the random variable is transformed into a standard normal and you can find probabilities using the standard normal table. For instance, assume U.S. adult heights and weights are both normally distributed The random variable of a standard normal distribution is considered as a standard score or z-score. Each normal random variable such as X can easily be converted into a z-score using the normal distribution z formula. z = (X − μ) σ In the above normal distribution z formula

Normal Distribution Formula The Gaussian distribution is defined by two parameters, the mean and the variance. When we want to express that a random variable X is normally distributed, we usually denote it as follows. X \sim N (\mu, \sigma^2) X ∼ N (μ,σ2 -score transformation formula is . Z = X. − µ σ / n ~ N (0, 1). Example: Suppose that the ages of a certain population are normally distributed, X. with mean µ = 27.0 years, and standard deviation σ = 12.0 years, i.e., X ~ N (27, 12). Sampling Distribution of a Normal Variable . Given a random variable . Suppose that the X populatio * Z = (x - μ) / σ Where*, Z = Standardized Random Variable x = The Value that is being Standardized μ = Mean of the Distribution σ = Standard Deviation of the Distribution Enter the value that is being standardized (x), Mean of the Distribution and Standard Deviation of the Distribution (σ) to calculate the standardized random variable 1. We first convert the problem into an equivalent one dealing with a normal variable measured in standardized deviation units, called a standardized normal variable. To do this, if X ∼ N(µ, σ5), then N(0, 1) X - Z = ~ σ µ 2. A table of standardized normal values (Appendix E, Table I) can then b Let a set of n random variables X=(XI,..., X) have a joint CDF=Fx(x 1, x2,..., x,), which can be given in terms of a sequence of conditional distribution functions: Fx( X) = Fx,(xl)Fx2 ix,(X2) --- Fxolx,. x ,(x

* The NORMINV formula is what is capable of providing us a random set of numbers in a normally distributed fashion*. The syntax for the formula is below: = NORMINV (Probability, Mean, Standard Deviation) The key to creating a random normal distribution is nesting the RAND formula inside of the NORMINV formula for the probability inpu The general **formula** for variance decomposition or the law of total variance is: If and are two **random** **variables**, and the variance of exists, then Var [ X ] = E ( Var [ X ∣ Y ] ) + Var ( E [ X ∣ Y ] ) . {\displaystyle \operatorname {Var} [X]=\operatorname {E} (\operatorname {Var} [X\mid Y])+\operatorname {Var} (\operatorname {E} [X\mid Y]). A real random vector = (, ,) is called a standard normal random vector if all of its components are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if (,) for all The standard normal probability distribution has mean equal to 40, whereas the value of random variable x is 80 and the z-statistic is equal to 1.8 then the standard deviation of standard normal probability distribution is

To compute for standard normal variable, three essential parameters are needed and these parameters are value (x), mean (μ) and standard deviation (σ). The formula for calculating standard normal variable: z = (x - μ) ⁄ Standard Normal Distribution Formula Standard Normal Distribution is a random variable which is calculated by subtracting the mean of the distribution from the value being standardized and then dividing the difference by the standard deviation of the distribution. The Formula of Standard Normal Distribution is shown below: Z = (X - μ) / In the case of a random variable which has distribution having a discrete component at a value , P ( X = b ) = F X ( b ) − lim x → b − F X ( x ) . {\displaystyle \operatorname {P} (X=b)=F_{X}(b)-\lim _{x\to b^{-}}F_{X}(x). The formula for the variance of a random variable is given by; Var(X) = σ 2 = E(X 2) - [E(X)] 2. where E(X 2) = ∑X 2 P and E(X) = ∑ XP. Functions of Random Variables. Let the random variable X assume the values x 1, x 2, with corresponding probability P (x 1), P (x 2), then the expected value of the random variable is given by

- A standard normal distribution has a mean of 0 and a standard deviation of 1. Standardized normal distribution formula is mentioned below. To compute the values from a standard normal distribution, subtract the mean of the distribution from the value that is being standardized
- This is also called a change of variable and is in practice used to generate a random variable of arbitrary shape f g(X) = f Y using a known (for instance, uniform) random number generator. It is tempting to think that in order to find the expected value E ( g ( X )), one must first find the probability density f g ( X ) of the new random variable Y = g ( X )
- The formulas for the mean and standard deviation are \(\mu = np\) and \(\sigma = \sqrt{npq}\). The mean is 159 and the standard deviation is 8.6447. The random variable for the normal distribution is \(X\). \(Y \sim N(159, 8.6447)\). See The Normal Distribution for help with calculator instructions
- Let us say, f(x) is the probability density function and X is the random variable. Hence, it defines a function which is integrated between the range or interval (x to x + dx), giving the probability of random variable X, by considering the values between x and x+dx. f(x) ≥ 0 ∀ x ϵ (−∞,+∞) And -∞ ∫ +∞ f(x) = 1. Normal Distribution Formula
- Probability Distributions of Discrete Random Variables. A typical example for a discrete random variable \(D\) is the result of a dice roll: in terms of a random experiment this is nothing but randomly selecting a sample of size \(1\) from a set of numbers which are mutually exclusive outcomes. Here, the sample space is \(\{1,2,3,4,5,6\}\) and we can think of many different events, e.g.
- Definition: standard normal random variable. A standard normal random variable is a normally distributed random variable with mean \(\mu =0\) and standard deviation \(\sigma =1\). It will always be denoted by the letter \(Z\)
- This random variable is normally distributed with mean 0 and standard deviation of 1. In Figure 1 I have marked the middle of both axis to make it clear that uniform random variables between 0 and 0.5 would map to normal random variables that are negative. The uniforms above 0.5 map to positive r.vs. Figure

One straightforward way to simulate a binomial random variable X X is to compute the sum of n n independent 0−1 random variables, each of which takes on the value 1 with probability p p. This method requires n n calls to a random number generator to obtain one value of the random variable any random variable, no matter what the values of and : for example, thenormal .5 probability that normal random variable takes on a value between one standardany deviation of its mean is 0.6827¸ Foot length (in inches) of a randomly chosen adult male is a normal random variable with a mean of 11 and standard deviation of 1.5. So X = foot length (inches). (a) Suppose that an XL sock is designed to fit the largest 30% of men's feet Let's start with a random variable X that has a normal distribution with mean = 10 and standard deviation = 2. Let's practice our new notation. Here we would write μ = 10 and σ = 2 . The normal curve for X is shown below

- A standard normal random variable is a normally distributed random variable with mean \(\mu =0\) and standard deviation \(\sigma =1\). It will always be denoted by the letter \(Z\). The density function for a standard normal random variable is shown in Figure \(\PageIndex{1}\)
- The formula to calculate standardized normal random variable is. This is a Most important question of gk exam. Question is : The formula to calculate standardized normal random variable is , Options is : 1. x - μ ⁄ σ, 2. x + μ ⁄ σ, 3.x - σ ⁄ μ, 4. x + σ ⁄ μ, 5. NULL
- Since Z1 will have a mean of 0 and standard deviation of 1, we can transform Z1 to a new random variable X=Z1*σ+μ to get a normal distribution with mean μ and standard deviation σ. = SQRT ( -2 * LN ( RAND ())) * COS ( 2 * PI () * RAND ()) * StdDev + Mean. Mean - This is the mean of the normal distribution
- Random variable: X = $ amount obtained Mean µ X = E [X] = (.5)(0) + (.3)(10) + (.2)(20) = $7.00 Variance 2 σ X = E [ (X - µ X) 2] = (.5)(−7) 2 + (.3)(3) 2 + (.2)(13) 2 = 61 Standard deviation σ X = $7.81 x f (x) = P (X = x) 0 .

The formula of calculating the expected value of random variable x of gamma distribution is as The demand of products per day for three days are 21, 19, 22 units and their respective probabilities are 0.29, 0.40, 0.35 Proof of Normal Distribution Formulas By . E-Pandu.Com. Tuesday, August 6, 2019 Add Comment Edit. We assume that X is a normal random variable or X is normally distributed,. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchang

numpy.random.normal ¶ numpy.random.normal(loc=0.0, scale=1.0, size=None) ¶ Draw random samples from a normal (Gaussian) distribution To standardize a normally distributed random variable, we need to calculate its Z score. The Z-score is calculates using two steps: (1) The mean of X is subtracted from X (2) Then divided that by the standard deviation of X. All possible observations are adjusted using this procedure to achieve a standard normal random variable, Z Then, the random variable defined as: has a normal distribution with mean and variance. Proof. First of all, we need to use the fact that mutually independent normal random variables are jointly normal: the random vector defined as has a multivariate normal distribution with mean and covariance matrix A Random Variable is a variable whose possible values are numerical outcomes of a random experiment. The Mean (Expected Value) is: μ = Σxp; The Variance is: Var(X) = Σx 2 p − μ 2; The Standard Deviation is: σ = √Var(X E e t Y − μ σ = E e t Y σ e − t μ σ. Note that e − t μ σ is a constant and it can be pulled out of the expectation. Now E e t Y σ is nothing but E e s Y where s = t σ. Use the formula you have for M Y (s) to finish

How to calculate the standard normal distribution . First, determine the normal random variable. Using the information provided or the formula Y = { 1/[ σ * sqrt(2π) ] } * e-(x - μ) 2 /2σ 2, determine the normal random variable. Determine the average. Calculate the mean or average of the data set. Determine the standard deviatio * Dividing by the standard deviation lets the variance of your new random variable be 1*. That is what the standardizing a random variable means. You simply let the mean and variance of your random variable be 0 and 1, respectivel We now turn to a continuous random variable, which we will denote by X. We will let the probability density function of X be given by the function f (x). The expected value of X is given by the formula: E (X) = ∫ x f (x) d x

Having the normally distributed Random variables. We can normalize it and use table values in order to calculate probability of some event. The standardization takes formula The normal distribution is an essential statistical concept as most of the random variables in finance follow such a curve. It plays an important part in constructing portfolios. Apart from finance, a lot of real-life parameters are found to be following such a distribution Normal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that arise in practice, such as the heights or weights of people, the total annual sales of a rm, exam scores etc. Also, it is important for th

The value of the normal random variable is 365 days. The mean is equal to 300 days. The standard deviation is equal to 50 days. Therefore, P (x < 365) = 0.903 Random Variables can be either Discrete or Continuous: Discrete Data can only take certain values (such as 1,2,3,4,5) Continuous Data can take any value within a range (such as a person's height) All our examples have been Discrete. Learn more at Continuous Random Variables The standard normal distribution is symmetric and has mean 0. 3.2 Properties of E(X) The properties of E(X) for continuous random variables are the same as for discrete ones: 1. If Xand Y are random variables on a sample space then E(X+ Y) = E(X) + E(Y): (linearity I) 2. If aand bare constants then E(aX+ b) = aE(X) + b: (linearity II) Example 5

- A vector-valued random variable X = X1 ··· Xn T is said to have a multivariate normal (or Gaussian) distribution with mean µ ∈ Rnn ++ 1 if its probability density function2 is given by p(x;µ,Σ) = 1 (2π)n/2|Σ|1/2 exp − 1 2 (x−µ)TΣ−1(x−µ) . of their basic properties. 1 Relationship to univariate Gaussians Recall that the density function of a univariate normal (or Gaussian) distribution is given b
- There is no simple formula for the distribution function of a standard normal random variable because the integral cannot be expressed in terms of elementary functions. Therefore, it is usually necessary to resort to special tables or computer algorithms to compute the values of
- The Normal Distribution The normal distribution plays an important role in the practice of risk management. There are many reasons for this. It is a relatively simple and tractable model that seems to capture adequately important aspects of many random variables. Of course, it has its limitations, whic
- 2 The Bivariate
**Normal**Distribution has a**normal**distribution. The reason is that if we have X = aU + bV and Y = cU +dV for some independent**normal****random****variables**U and V,then Z = s1(aU +bV)+s2(cU +dV)=(as1 +cs2)U +(bs1 +ds2)V. Thus, Z is the sum of the independent**normal****random****variables**(as1 + cs2)U and (bs1 +ds2)V, and is therefore normal.A very important property of jointly**normal****random**. - If we have 2 normal, uncorrelated random variables $X_1, X_2$ then we can create 2 correlated random variables with the formula $Y=\rho X_1+ \sqrt{1-\rho^2} X_2$ and then $Y$ will have a correlation $\rho$ with $X_1$
- 20.1 - Two Continuous Random Variables; 20.2 - Conditional Distributions for Continuous Random Variables; Lesson 21: Bivariate Normal Distributions. 21.1 - Conditional Distribution of Y Given X; 21.2 - Joint P.D.F. of X and Y; Section 5: Distributions of Functions of Random Variables. Lesson 22: Functions of One Random Variable

Where, X = Random variable; Examples of Normal Distribution in Statistics. Let's discuss the following examples. Example #1. Suppose a company has 10000 employees and multiple salaries structure as per the job role in which employee works Let X be a normal random variable with given mean and variance. This means that the PDF of X takes the familiar form. We consider random variable Y, which is a linear function of X. And to avoid trivialities, we assume that a is different than zero. We will just use the formula that we have already developed

A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment's outcomes. A random variable can be either discrete (having specific values) or. Normal Distribution Summary. Normal distribution or Gaussian Distribution is a statistical distribution which is widely used in the analytical industry and have a general graphical representation as a bell-shaped curve which has exactly half of the observations at the right hand side of Mean/Median/Mode and exactly half of them on the left hand side of Mean/Median/Mode Standard deviation definition formula. The standard deviation is the square root of the variance of random variable X, with mean value of μ. From the definition of the standard deviation we can get. Standard deviation of continuous random variable. For continuous random variable with mean value μ and probability density function f(x): o Log-normal distribution. by Marco Taboga, PhD. A random variable is said to have a log-normal distribution if its natural logarithm has a normal distribution.In other words, the exponential of a normal random variable has a log-normal distribution One day it just comes to your mind to count the number of cars passing through your house. The number of these cars can be anything starting from zero but it will be finite. This is the basic concept of random variables and its probability distribution. Here the random variable is the number of the cars passing

We will show this in the special case that both random variables are standard normal. The general case can be done in the same way, but the calculation is messier. Another way to show the general result is given in Example 10.17. Suppose X and Y are two independent random variables, each with the standard normal density (see Example 5.8). We hav The normal distribution is defined by the following equation: Normal equation.The value of the random variable Y is:. Y = { 1/[ σ * sqrt(2π) ] } * e-(x - μ) 2 /2σ 2. where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.14159, and e is approximately 2.71828.. In this equation, the random variable X is called a normal random variable The z-score statistics. Given any normal variable, x with mean = and standard deviation = , we define the z-score or standard score (the z value of the standard normal distribution) as: So given any value of a normal random variable, x, and its related mean and standard deviation, then the x value can be converted to a standard normal variable, z by the formula above and be used to find. But in statistics, it is normal to use an X to denote a random variable. The random variable takes on different values depending on the situation. Each value of the random variable has a.

Multivariate normal distribution. by Marco Taboga, PhD. The multivariate normal (MV-N) distribution is a multivariate generalization of the one-dimensional normal distribution.In its simplest form, which is called the standard MV-N distribution, it describes the joint distribution of a random vector whose entries are mutually independent univariate normal random variables, all having zero. randn in matlab produces normal distributed random variables W with zero mean and unit variance. To change the mean and variance to be the random variable X (with custom mean and variance), follow this equation: X = mean + standard_deviation*W Please be aware of that standard_deviation is square root of variance

- what I want to do in this video is build up some tools in our toolkit for dealing with sums and differences of random variables so let's say that we have two random variables x and y and they are completely independent they are independent independent random variables random variables and I'm just going to go over a little bit of notation here if we wanted to know the expected or if we talked.
- Your intuition is correct - the marginal distribution of a normal random variable with a normal mean is indeed normal. To see this, we first re-frame the joint distribution as a product of normal densities by completing the square
- I discuss standardizing normally distributed random variables (turning variables with a normal distribution into something that has a standard normal distrib..
- The number of correct answers X is a binomial random variable with n = 100 and p = 0.25. Thus this random variable has mean of 100(0.25) = 25 and a standard deviation of (100(0.25)(0.75)) 0.5 = 4.33. A normal distribution with mean 25 and standard deviation of 4.33 will work to approximate this binomial distribution
- Well, first we'll work on the probability distribution of a linear combination of independent normal random variables \(X_1, X_2, \ldots, X_n\). On the next page, we'll tackle the sample mean! Theorem
- e probabilities for the sample [
- Chapter 6: Random Variables and the Normal Distribution Random Variable Formulas for the Variance and Standard Deviation of a Discrete Random Variable 2 2 2 Definition Formulas X P X X P X 2 2 2 22 Computational Formulas X P X X P X. Example x P(x) 0 0.0625 0 0

the tables for the standard normal random variable Z, given above, to calculate the probability. Property If Xis a normal random variable with mean and standard deviation ˙, then the random variable Z de ned by the formula (Note this is the Z-score of X): Z= X ˙ has a standard normal distribution

- Useful formula for moment computation of normal random variables with nonzero means Abstract: It is well known that higher order moments of normal random variables (RV) with zero means can be expressed by terms of second-order moments
- Random number distribution that produces floating-point values according to a normal distribution, which is described by the following probability density function: This distribution produces random numbers around the distribution mean (μ) with a specific standard deviation (σ). The normal distribution is a common distribution used for many kind of processes, since it is the distribution.
- The randn function returns a sample of random numbers from a normal distribution with mean 0 and variance 1. The general theory of random variables states that if x is a random variable whose mean is μ x and variance is σ x 2 , then the random variable, y , defined by y = a x + b , where a and b are constants, has mean μ y = a μ x + b and variance σ y 2 = a 2 σ x 2
- • Random Variables. Random Variables! -1 0 1 A rv is any rule (i.e., function) that associates a number with each outcome in the sample space. Two Types of Random Variables •A discrete random variable has a countable number of possible values Normal Distribution. Binomial Distributio
- remain normal. ! Meansof normal variables are normally distributed. Central Limit Theorem:Means of non-normal variables are approximately normally distributed. ! Hypothesis ofElementary Errors: If random variation is the sum of many small random effects, a normal distribution must be the result. Regression modelsassume normally.
- normal random variable z values using the formula 3 From the Tables find the from BUSINESS ST1002 at Trinity College Dubli
- about normal and related distributions. 2. Properties of the Normal and Lognormal Distributions First of all, a random variable Z is called standard normal (or N.0;1/, for short), if its density function f Z.z/ is given by the standard normal density function ˚.z/:DDe z2 =2 p 2ˇ. The function 8.z/:D R z 1 ˚.u/du denotes th

Statistics and ProbabilityQuarter 3 - Module All random variables (discrete and continuous) have a cumulative distribution function. It is a function giving the probability that the random variable X is less than or equal to x, for every value x. Formally, the cumulative distribution function F (x) is defined to be: F (x) = P (X<=x) for. -infinity < x < infinity This approach for generation of random variables works well provided that the CDF of the desired distribution is invertible. One notable exception where this approach will be difficult is the Gaussian random variable. Suppose, for example, we wanted to transform a uniform random variable, X, into a standard normal random variable, Y

A random variable is a variable that is subject to randomness, which means it can take on different values. As in basic math, variables represent something, and we can denote them with an x or a y. Simple Example. The random variable X is given by the following PDF. Check that this is a valid PDF and calculate the standard deviation of X.. Solution Part 1. To verify that f(x) is a valid PDF, we must check that it is everywhere nonnegative and that it integrates to 1.. We see that 2(1-x) = 2 - 2x ≥ 0 precisely when x ≤ 1; thus f(x) is everywhere nonnegative * Normal random variables A random variable X is said to be normally distributed with mean µ and variance σ2 if its probability density function (pdf) is f X(x) = 1 √ 2πσ exp − (x−µ)2 2σ2 , −∞ < x < ∞*. (1.1) Whenever there is no possible confusion between the random variable X and th We use the Normal Distribution Calculator to compute both probabilities on the right side of the above equation. To compute P ( X < 110 ), we enter the following inputs into the calculator: The value of the normal random variable is 110, the mean is 100, and the standard deviation is 10. We find that P ( X < 110 ) is 0.84 numpy.random.normal¶ numpy.random.normal (loc=0.0, scale=1.0, size=None) ¶ Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently , is often called the bell curve because of its characteristic shape (see the example below)

- Cumulative Normal Probabilities: In Excel, the above cumulative probability can be calculated by the following formula: Conversion from X to Z and vice versa: Given a Normal random variable X with a mean µ and a standard deviation σ, the following formula will convert (transform) it into a standard normal random variable Z: z = (x - µ) / σ (I) Conversely, the following formula converts the.
- The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There are two major reasons to employ such a correction. First, recall that a discrete random variable can only take on only speciﬁed values
- In Section 3.10.3, we saw that the specific form of quadratic polynomial of a joint standard normal random vector has a chi-squared distribution. Generalizing this, we shall demonstrate that any quadratic polynomial of any joint-normal random vector can be expressed as a linear polynomial of independent chi-squared and normal random variables
- 5 Calculations with General Normal Random Variable via the Normal Table Given x-value, calculate probability Given probability, calculate x-value Donglei Du (UNB) ADM 2623: Business Statistics 30 / 53. Given x-value, calculate probability Example: Given a normal random variable X˘N(50;82), calculat
- expressing the random variable T as a function of the random variables and S. Weʼll first discuss the t-statistic in the case where our underlying random variable Y is normal, then extend to the more general situation stated in Chapter 23. 2. The case of Y normal. For Y normal, we will use the following theorem

random variables implies that the events fX •5gand f5Y3 C7Y2 ¡2Y2 C11 ‚0gare independent, and that the events fX evengand f7 •Y •18gare independent, and so on. In-dependence of the random variables also implies independence of functions of those random variables. For example, sin.X/must be independent of exp.1 Ccosh.Y2 ¡3Y//, and so on itive random variables that are independent of the nonnegative integer-valued ran-dom variable N+ The random variable SN 5 (i51 N X i is called a compound random variable+ In Section 2, we give a simple probabilistic proof of an identity concern-ing the expected value of a function of a compound random variable; when the X For any functions g and h (because if X and Y are independent, so are g (X) and h (y)). Now, at last, we're ready to tackle the variance of X + Y. We start by expanding the definition of variance: By (2): Now, note that the random variables and are independent, so: But using (2) again: is obviously just , therefore the above reduces to 0 where [z.sub. [alpha]/2] is the positive value that the standard normal random variable exceeds with a probability of [alpha]/2. On the possibility of a private crop insurance market: a spatial statistics approach. For normal random variables, X can be transformed into standard normal random variables u through a linear transformation, as follows

We use the formula for Z transformation: Normally distributed Random Variable = 10 = 2 Standard Normal Distribution = 1 = 0 5. The mean of the original distribution is 10 and it translates to: Normally distributed Random Variable = 10 = 2 Standard Normal Distribution = 1 = 0 6 A random variable is defined as a variable whose value describes the outcome of a random event. This random event can be flipping a coin, where the corresponding random variable takes a value of either head or tail; Or, it can be measuring the height of an adult male, where the corresponding random variable takes a value anywhere between 0.54m ( shortest man on earth ) and 2.51m ( tallest man. Conditional expectation and covariance of two sets of normal random variables. E.36.26 Conditional expectation and covariance of two sets of normal random variables Consider a (¯n+¯k)-dimensional normal variable (XZ)∼N((μXμZ),(σ2X.. The two parameters that are needed to define a normal are: , , this explanation will be developed in chapter 6. In general if you have a Normal random variable with parameters and , we need to standardize it, because the probabilities cannot be computed from a closed form formula, this is done by standardizing, sa The normal table outlines the precise behavior of the standard normal random variable Z, the number of standard deviations a normal value x is below or above its mean. The normal table provides probabilities that a standardized normal random variable Z would take a value less than or equal to a particular value z*