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Mathematics LibreTexts

8.1: Normal Random Variables

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Section 1: Introduction

Definition: Normal probability density curve

A normal probability density curve with parameters μ and σ is a bell-shaped curve that satisfies the following properties:

  1. Has the peak at μ and symmetric about μ.
  2. Extends indefinitely in both directions, approaching but never touching the horizontal axis.
  3. The Empirical rule holds that is:
    1. ~68% of the area under the curve is between μ1σ and μ+1σ;
    2. ~95% of the area under the curve is between μ2σ and μ+2σ;
    3. ~99.7% of the area under the curve is between μ3σ and μ+3σ.

The description of the curve should sound familiar as we have seen it before!

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Definition: Normal random variable with parameters μ and σ

We define a normal random variable with parameters μ and σ as a continuous random variable with a normal probability density curve with parameters μ and σ. We denote such a random variable in the following way:

XN(μ,σ)

Definition: Parameters of a normal distribution

The values, μ and σ, are called the parameters of a normal distribution and each pair of such values determines a unique shape of the probability density curve.

Example 8.1.1.1.1

When μ=50 and σ=10, we get the following curve:

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Example 8.1.1.1.2

When μ=8 and σ=2, we get the following curve:

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Try It Yourself! 8.1.1.1

Let’s make sure we understand the properties of normal probability density curves and use the empirical rules to find some probabilities for any XN(μ,σ).

  • To find the probability that X is less than μ we will look for the area under the curve to the left of μ. This region’s area can be found by adding the areas of included regions together to get P(X<μ)=0.50 or 50%.
  • To find the probability that X is between μσ and μ+σ we will look for the area under the curve between μσ and μ+σ. This region’s area can be found by adding the areas of included regions together to get P(μσ<X<μ+σ)=0.68 or 68%.
  • To find the probability that X is between μ and μ+2σ we will look for the area under the curve between μ and μ+2σ. This region’s area can be found by adding the areas of included regions together to get P(μ<X<μ+2σ)=0.475 or 47.5%.
  • To find the probability that X is between μσ and μ+2σ we will look for the area under the curve between μσ and μ+2σ. This region’s area can be found by adding the areas of included regions together to get P(μσ<X<μ+2σ)=0.815 or 81.5%.
  • To find the probability that X is greater than μ+σ we will look for the area under the curve to the right of μ+σ. This region’s area can be found by adding the areas of included regions together to get P(X>μ+σ)=0.16 or 16%.
  • To find the probability that X is less than μ2σ we will look for the area under the curve to the left of μ2σ. This region’s area can be found by adding the areas of included regions together to get P(X<μ2σ)=0.025 or 2.5%.
  • To find the probability that X is greater than μ+4σ we will look for the area under the curve to the right of μ+4σ. This region’s area is essentially equal to 0 so the probability is nearly 0%, that is P(X>μ+4σ)=0.
Example 8.1.1.2.1

For XN(μ=50,σ=10), let’s find the following probabilities:

  • To find the probability that X is less than 50 we will look for the area under the curve to the left of 50. This region’s area can be found by adding the areas of included regions together to get P(X<50)=0.50 or 50%.
  • To find the probability that X is between 40 and 70 we will look for the area under the curve between 40 and 70. This region’s area can be found by adding the areas of included regions together to get P(40<X<70)=0.815 or 81.5%.
Example 8.1.1.2.2

For XN(μ=8,σ=2), let’s find the following probabilities:

  • To find the probability that X is between 8 and 12 we will look for the area under the curve between 8 and 12. This region’s area can be found by adding the areas of included regions together to get P(8<X<12)=0.475 or 47.5%.
  • To find the probability that X is between 6 and 12 we will look for the area under the curve between 6 and 12. This region’s area can be found by adding the areas of included regions together to get P(6<X<12)=0.815 or 81.5%.
Try It Yourself! 8.1.1.2

Section 2: Working with Normal Random Variables

For a normal random variable with parameters μ and σ, we want to be able to find the probabilities that involve the values a, b, and c that are not covered by the Empirical rule:

  • P(X<c)
  • P(X>c)
  • P(a<X<b)

But first let’s make sure we understand the questions!

  • To find the probability that X is less than c, we will look for the area under the curve to the left of c.
  • To find the probability that X is greater than c, we will look for the area under the curve to the right of c.
  • To find the probability that X is between a and b, we will look for the area under the curve to the right of a and to the left of b.

While it is clear what the regions’ areas look like they cannot be found by using the empirical rule. So how do we find the probabilities that involve values that are not covered by the Empirical rule such as P(X<c), P(X>c), P(a<X<b)?

For XN(μ=8,σ=2), let's find the following probabilities:

  • P(X<10.5)
  • P(X>9.3)
  • P(7.1<X<8.7)

Luckily, there are ways to use technology to answer these questions:

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However, it is worth learning how to perform this task by hand. To be able to answer the same questions manually we have to review the concept of a z-score.

Definition: z-score

The z-score of an observation xi is computed by using the formula:

zscore=xiμσ

Example 8.1.2.1

Let’s find some z-scores for a normal random variable X with μ=8 and σ=2:

  • the z-score of x1=7.1 is 7.182=0.45
  • the z-score of x2=8.7 is 8.782=0.35
  • the z-score of x3=9.3 is 9.382=0.65
  • the z-score of x4=10.5 is 10.582=1.25
Try It Yourself! 8.1.2.1

If instead of observing a normal variable with parameters μ and σ we consider the z-scores of the observed values. Then it is not too hard to observe that the z-scores also have a normal distribution with parameters 0 and 1 which we call the standard normal distribution. In other words, the new random variable Xμσ has the standard normal distribution, therefore:

Xμσ=ZN(0,1)

By the way, the fact that the standard normal distribution is called Z-distribution explains the origin of the name for a term “a z-score”.

Definition: Standardizing a random variable

The process of converting the original normal random variable to Z is called standardizing the random variable.
Most importantly, the area between any two values a and b under the normal curve is the same as the area between the z-scores of the a and b under the z-curve:

P(a<X<b)=P(za<Z<zb)

Example 8.1.2.2

For XN(ν=8,σ=2), find the following probabilities:

  • P(X<10.5)=P(Z<10.582)=P(Z<1.25)=0.8944
  • P(X>9.3)=P(Z>9.382)=P(Z>0.65)=0.2578
  • P(7.1<X<8.7)=P(7.182<Z<8.782)=P(0.45<Z<0.35)=P(Z<0.35)P(Z<0.45)=0.63680.3264=0.3104
Try It Yourself! 8.1.2.2.1
Try It Yourself! 8.1.2.2.2
Try It Yourself! 8.1.2.2.3

Similarly, we can use the standardization process for any other normal random variable.

Example 8.1.2.3

For X N(μ=50,σ=10), find the following probabilities:

  • P(X<58)=P(Z<585010)=P(Z<0.8)=0.7881
  • P(X>32)=P(Z>325010)=P(Z>1.8)=0.9641
  • P(47<X<69)=P(475010<Z<695010)=P(0.3<Z<1.9)=0.589

Of course, every result obtained manually can be confirmed using technology:

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Next, we will discuss how to find xα. To find xα means to find the value the area to the right of which is equal to α. In other words, we are looking for xα such that

P(X>xα)=α

Standardizing the left-hand side of the equation gives us

P(X>xα)=P(Z>xαμσ)=α

which implies that

xαμσ=zα

therefore

xα=μ+zασ

Example 8.1.2.4.1

For a normal random variable with parameters μ=8 and σ=2, let’s find x0.38 by using the above formula.

x0.38=μ+z0.38σ

Now that we related x_{0.38} to z_{0.38} we use the z-table as we have done it before to identify α=0.38 and 1α=0.62 to obtain the z0.38=0.31. Now we can compute the desired x0.38 by finishing the calculations:

x0.38=μ+z0.38σ=8+0.312=8.62

Note that the following two statements are true by the definition of the alpha notation:

P(X<8.62)=0.62 and P(X>8.62)=0.38

Example 8.1.2.4.2

For a normal random variable with parameters μ=50 and σ=10, let’s find x0.10 by using the above formula.

x0.10=μ+z0.10σ

Now that we related x0.10 to z0.10 we use the z-table as we have done it before to identify α=0.10 and 1α=0.90 to obtain the z0.10=1.28. Now we can compute the desired x0.10 by finishing the calculations:

x0.10=μ+z0.10σ=50+1.2810=62.8

Try It Yourself! 8.1.2.4.1
Try It Yourself! 8.1.2.4.2

Now that we know how to find the probabilities and xαs let’s discuss the parameters of a normal distribution. It is not a coincidence that the two parameters, μ and σ, for a normal random variable are labeled similar to the mean and the standard deviation of the data set. The meaning and the interpretation of the mean and standard deviation for a continuous random variable is the same as for discrete random variables. That is, the expectation (or the mean) of a random variable is a measure of the center of the distribution and the variance (and standard deviation) is a measure of the spread of the random variable.

Theorem: Mean, variance, and standard deviation of a Normal RV

For XN(μ,σ):

E[N(μ,σ)]=μ

VAR[N(μ,σ)]=σ2

SD[N(μ,σ)]=σ

Example 8.1.2.5

For example, for XN(10,2) by the three standard deviations rule, we may conclude that:

Range

Numerical Range

Interpretation

within 2σX from μX

between 6 and 14

“expected”

more than 2σX from μX

less than 6 or more than 14

“unusual”

more than 3σX from μX

less than 4 or more than 16

“abnormal”

Try It Yourself! 8.1.2.5

This page titled 8.1: Normal Random Variables is shared under a not declared license and was authored, remixed, and/or curated by Anton Butenko.

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