Homework 12

Additional Directions

For each of the following problems:

1. Sketch the probability density function of the Gaussian random variable (i.e. get the general shape and placement of the probability density function correct).
2. Indicate the area under the curve that you are computing to answer the given probability query.
3. Use R to compute the probability, and give the R command you used.

Example: If \(X \sim N(1, 2^{2})\), and you are asked to compute \(P(0 \leq X \leq 2)\), you should sketch:

as well as state and find \[ \begin{aligned} P(0 \leq X \leq 2) &= P(X \leq 2) - P(X \leq 0) \\ &= \texttt{pnorm(2, mean = 1, sd = 2) - pnorm(0, mean = 1, sd = 2)} \end{aligned} \]

Notice: I have marked the horizontal axis with \(\mu \pm k \sigma\) for \(k = 0, 1, 2, 3\).

Hint: You may find this Shiny demo useful as you get familiar with the Gaussian curve and using R to compute Gaussian probabilities. This Shiny app will generate practice problems you can work through and then check your work.

Problems

Section 4.3:

• 39
• 43; for this problem use pnorm with arguments mean = 0 and sd = 1. This means you need to standardize \(X\) first.
• 44

Read pages 178-180 (up to but not including "Nonstandard Normal Distributions") and page 183 (up to but not including "The Normal Distribution and Discrete Populations") in Devore & Berk about percentiles of the Gaussian (normal) distribution, and solve the following problems.

Section 4.3:

• 41
• 42
• 52; Note: This is a challenging problem. Please see me during office hours or email me if you would like any hints after attempting this problem.