### Chapter 4. Probability Distributions: Continuous Probability Distributions

### The Normal Probability Distribution

Normal Probability Distribution

If #X# is a value sampled from a *population *having a #N(\mu,\sigma)# distribution, then #X# is a random variable having a #N(\mu, \sigma)# *probability distribution*.

Then for any number #k#:

- #\mathbb{P}(X \leq k)# is the area to the left of #k#
- #\mathbb{P}(X \gt k)# is the area to the right of #k#

Computation of Normal Probabilities with Statistical Software

Let #X# be a *normal *random variable with parameters #\mu# and #\sigma#.

To calculate *cumulative probabilities *for a *normal *distribution in Excel, make use of the following function:

NORM.DIST(x, mean, standard_dev, cumulative)

x: The value at which you wish to evaluate the distribution function.mean: The mean of the distribution.standard_dev: The standard deviation of the distribution.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution function, #\mathbb{P}(X \leq x)#
- FALSE - uses the probability density function, #\mathbb{P}(X = x)#

To calculate *cumulative probabilities *for a *normal *distribution in R, make use of the following function:

pnorm(q, mean, sd, lower.tail)

q: The value at which you wish to evaluate the distribution function.mean: The mean of the distribution.sd: The standard deviation of the distribution.lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

There are a number of different ways we can calculate #\mathbb{P}(X \lt 75)#. Click on one of the panels to toggle a specific solution.

Since the normal distribution is a *continuous *distribution, it is true that:

\[\mathbb{P}(X \lt 75)=\mathbb{P}(X \leq 75)\]

To calculate #\mathbb{P}(X \leq 75)# in Excel, make use of the following function:

NORM.DIST(x, mean, standard_dev, cumulative)

x: The value at which you wish to evaluate the distribution function.mean: The mean of the distribution.standard_dev: The standard deviation of the distribution.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution function, #\mathbb{P}(X \leq x)#
- FALSE - uses the probability density function

Thus, to calculate #\mathbb{P}(X \leq 75)#, run the following command:

\[= \text{NORM.DIST}(75, 100, 15, \text{TRUE})\]

This gives:

\[\mathbb{P}(X \lt 75) = \mathbb{P}(X \leq 75) =0.048\]

Since the normal distribution is a *continuous *distribution, it is true that:

\[\mathbb{P}(X \lt 75)=\mathbb{P}(X \leq 75)\]

To calculate #\mathbb{P}(X \leq 75)# in R, make use of the following function:

Thus, to calculate #\mathbb{P}(X \leq 75)#, run the following command:

pnorm(q, mean, sd, lower.tail)

q: The value at which you wish to evaluate the distribution function.mean: The mean of the distribution.sd: The standard deviation of the distribution.lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

\[\text{pnorm}(q = 75, mean = 100, sd = 15, lower.tail = \text{TRUE})\]

This gives:

\[\mathbb{P}(X \lt 75) = \mathbb{P}(X \leq 75) =0.048\]

#\phantom{0}#

pth Quantile

The #\boldsymbol{p^{th}}# **quantile **of a probability distribution is the number #q# such that #\mathbb{P}(X\leq q)=p#.

Finding the #p^{th}# quantile for a given #p# in #(0,1)# is the *inverse *of finding a probability.

Finding the pth Quantile with Excel

Let #X# be a *normal *random variable with parameters #\mu# and #\sigma#.

To calculate the number #q# such that #\mathbb{P}(X \leq q)=p# in Excel, make use of the following function:

NORM.INV(probability, mean, standard_dev)

probability: A probability corresponding to the normal distribution.mean: The mean of the distribution.standard_dev: The standard deviation of the distribution.

To calculate the number #q# such that #\mathbb{P}(X \leq q)=p# in R, make use of the following function:

qnorm(p, mean, sd, lower.tail)

p: A probability corresponding to the normal distribution.mean: The mean of the distribution.sd: The standard deviation of the distribution.lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

Find the number #q# such that #\mathbb{P}(X \leq q) = 0.65#. Round your answer to #2# decimal places.

#q=113.85#

There are a number of different ways we can calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.65#. Click on one of the panels to toggle a specific solution.

To calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.65# in Excel, make use of the following function:

NORM.INV(probability, mean, standard_dev)

probability: A probability corresponding to the normal distribution.mean: The mean of the distribution.standard_dev: The standard deviation of the distribution.

Thus, to calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.65#, run the following command:

\[=\text{NORM.INV}(0.65, 110, 10)\]

This gives:

\[q = 113.85\]

To calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.65# in R, make use of the following function:

Thus, to calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.65#, run the following command:

qnorm(p, mean, sd, lower.tail)

p: A probability corresponding to the normal distribution.mean: The mean of the distribution.sd: The standard deviation of the distribution.lower.tail: If TRUE (default), probabilities are #\mathbb{P}(X \leq x)#, otherwise, #\mathbb{P}(X \gt x)#.

\[\text{qnorm}(p = 0.65, mean = 110, sd = 10, lower.tail = \text{TRUE})\]

This gives:

\[q = 113.85\]

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