stat310

# Homework 05

This homework follows the standard late penalty: 0% if in the stat310 mailbox by Thursday 16 Feb 4pm, 10% by 5pm the following day, 100% otherwise. Please read the syllabus for other homework policies.

1. (4 pts each) For each of the following random variables, identify the distribution that most closely matches the situation. Justify your choice and describe any assumptions that you made.

1. On average, flight from Houston (HOU) to Dallas (DAL) leaves every 60 minutes. Let $$X_{1}$$ be the amount of time I have to wait for the next airplane to come.

2. Despite what it says on the bottle, the amount of beer in a bottle actually varies a little. 12oz beer bottles have a mean volume of 11.8 oz and on variance on 0.7 oz. Let $$X_{2}$$ be the amount of beer in the bottle from this sample.

3. The registrar gets lazy and decides to use a random number generator to determine the threshold GPA for the President’s Honor Roll. Let $$X_{3}$$ be the minimum GPA to get you on the Honor Roll.

2. (2 pts each) For each of the following random variables, find the specified probability using the CDF.

1. $$X_{1} \sim Exp(\theta = 10)$$, $$P(10 < X_{1} < 100)$$

2. $$X_{2} \sim Gamma(\alpha = 1, \beta = 2)$$, $$P(1 < X_{2} < 5)$$

3. $$X_{3} \sim Normal(\mu = 0, \sigma^{2} = 10)$$, $$P(-10 < X_{3} < 10)$$. Don’t use wolfram alpha!

3. (4 pts each) Given that $$X \sim Exp(\theta)$$, find the pdfs of the following two transformations of X. Do they correspond to named distributions that we know about?

1. $$Y = X^{2}$$
2. $$Z = e^{X}$$
4. (Bonus 3pts) Show that if $$X \sim Normal(\mu, \sigma^2))$$ then $$Z = (X - \mu) / \sigma$$ has a standard normal distribution.