Chapter 35: Normal Random Variables

Meike Niederhausen and Nicky Wakim

2023-11-29

Learning Objectives

  1. Translate a word problem into probability within Normal RV

  2. Calculate probabilities within Normal RV using R

Properties of Normal RVs

  • No scenario description here because the Normal distribution is so universal

    • Central Limit Theorem (next class) makes it applicable to many types of events

  • Shorthand: \(X \sim \text{Normal}(\mu, \sigma^2)\)

\[ f_X(x) = \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)} \text{, for} -inf < x < inf \]

\[\text{E}(X) = \mu \] \[\text{Var}(X) = \sigma^2\]

Helpful R code

Let’s say we’re measuring the high temperature today. The average high temperature on this day across many, many years is 50 degrees with a standard deviation of 4 degrees.

  • If we want to know the probability that the high temperature is below 45 degrees:

    pnorm(q = 45, mean = 50, sd = 4)
    [1] 0.1056498

  • If we want to know the temoerature, say \(t\), where the probability of that the temperature is at \(t\) or lower is 0.35:

    qnorm(p = 0.35, mean = 50, sd = 4)
    [1] 48.45872

  • If we want to know the probability that the temperature is between 45 and 50 degrees:

    pnorm(q = 50, mean = 50, sd = 4) - pnorm(q = 45, mean = 50, sd = 4)
    [1] 0.3943502

  • If we want to sample 20 days’ temperature (over the years) from the distribution:

    rnorm(n = 20, mean = 50, sd = 4)
     [1] 47.44465 43.56419 56.41578 52.20924 53.92320 48.78870 46.04027 54.39277
 [9] 52.01942 50.05523 49.14754 47.46886 46.37231 51.44163 49.46695 47.14421
[17] 47.87237 46.76406 47.01549 50.22988

Movie night while studying

Example 1

Children’s movies run an average of 98 minutes with a standard deviation of 10 minutes. You check out a random movie from the library to entertain your kids so you can study for your test. Assume that your kids will be occupied for the entire length of the movie.

  1. What is the probability that your kids will be occupied for at least the 2 hours you would like to study?

  2. What is range for the bottom quartile (lowest 25%) of time they will be occupied?

Standard Normal Distribution

\[ Z \sim \text{Normal}(\mu = 0, \sigma^2 = 1)\]

  • Used to be more helpful when computing was not as advanced

    • Use tables of the standard normal

    • You can convert any normal distribution to a standard normal through transformation

  • \(Z = \dfrac{X - \mu_X}{\sigma_X}\)

    • Comes from \(X = \sigma_X Z + \mu_X\)

    • Since \(\sigma_X\) and \(\mu_X\) are constants, then \(E(X) = \mu_X\) and \(SD(X) = \sigma_X SD(Z) = \sigma_X\)

Chapter 35 Slides