4. Probability Distributions: Common 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#
Important Properties
For any continuous probability distribution, the following properties always hold:
- #\mathbb{P}(X=k)=0#
- #\mathbb{P}(X\leq k)=\mathbb{P}(X\lt k)#
- #\mathbb{P}(X\geq k)=\mathbb{P}(X\gt k)#
- #\mathbb{P}(X\gt k)= 1 - \mathbb{P}(X \leq k)#
- #\mathbb{P}(X\lt k)= 1 - \mathbb{P}(X \geq k)#
Additionally, for any #2# numbers #j# and #k# (with #j<k#), the following property hold:
- #\mathbb{P}(j \leq X \leq k) = \mathbb{P}(X \leq k) - \mathbb{P}(X \leq j)#
#\phantom{0}#
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
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 93)#. Click on one of the panels to toggle a specific solution.
Since the normal distribution is a continuous distribution, it is true that:
\[\mathbb{P}(X \lt 93)=\mathbb{P}(X \leq 93)\]
To calculate #\mathbb{P}(X \leq 93)# 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 93)#, run the following command:
\[= \text{NORM.DIST}(93, 100, 15, \text{TRUE})\]
This gives:
\[\mathbb{P}(X \lt 93) = \mathbb{P}(X \leq 93) =0.320\]
Since the normal distribution is a continuous distribution, it is true that:
\[\mathbb{P}(X \lt 93)=\mathbb{P}(X \leq 93)\]
To calculate #\mathbb{P}(X \leq 93)# in R, make use of the following function:
Thus, to calculate #\mathbb{P}(X \leq 93)#, 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)#.
\[\text{pnorm}(q = 93, mean = 100, sd = 15, lower.tail = \text{TRUE})\]
This gives:
\[\mathbb{P}(X \lt 93) = \mathbb{P}(X \leq 93) =0.320\]
#\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 Statistical Software
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) = 0.43#. Round your answer to #2# decimal places.
#q=138.24#
There are a number of different ways we can calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.43#. Click on one of the panels to toggle a specific solution.
To calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.43# 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) = 0.43#, run the following command:
\[=\text{NORM.INV}(0.43, 140, 10)\]
This gives:
\[q = 138.24\]
To calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.43# in R, make use of the following function:
Thus, to calculate the number #q# such that #\mathbb{P}(X \leq q) = 0.43#, 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)#.
\[\text{qnorm}(p = 0.43, mean = 140, sd = 10, lower.tail = \text{TRUE})\]
This gives:
\[q = 138.24\]
