NOTE: This homework is due on Tuesday, April 7. Please turn in your solutions on Gradescope. There is a chance that I will add a couple of problems on Thursday.


  1. Robin is teaching a huge statistics course. The score of a randomly selected student from this course can be modeled as \(X~N(\mu=84, \sigma^2 = 25)\).
  1. Shankar decides he is going to convert the scores to out of 50. So let \(Y = X/2\). What is the distribution of \(Y\).
  2. Let \(Y = (X − 84)/5\). What is the distribution of \(Y\).
  3. Let \(Y = 2X + 30\). What is the distribution of Y?



2. Let \(X\) be a random variable with pdf given by \[f(x) = \frac{1}{2}\exp(-|x|), \ \ \ \ -\infty<x<\infty.\] Compute the probabilities of the following events.

  1. \(\{|X|\leq 2\}\)
  2. \(\{|X|\leq 2 \textbf{ or } X\geq 0\}\)
  3. \(\{|X|\leq 2 \textbf{ or } X\leq -1\}\)
  4. \(\cup_{k=1}^\infty\{X=k\}\)


3. A candy bar company advertises the weight of the bars as 4 ounces. However, the weights of the bars are actually random variables assumed to be Normal with parameters \(\mu = 4.25\) and variance \(\sigma^2 = (0.1)^2 = 0.01\). The manufacturer wants to find an interval centered at 4.25 that will contain 95% of the weights. That is, they want to find \(a\) such that \[P(4.25-a \leq X \leq 4.25+a) = 0.95\] What is \(a\)?


4. Henry and Meghan move from the UK (where the temperature is measured in Celsius) to Chapel Hill (where, oddly, the temperature is still measured in Fahrenheit). Before leaving the UK, they are told that the temperature in Chapel Hill is a random variable \(X\) (measured in Celsius) that has a normal distribution \(N(25, 2^2)\). When the formerly royal couple arrive in Chapel Hill on June 1st, what is the probability that the temperature is more than 81.5 Fahrenheit?


5. Bazinga airlines knows that a passenger who books a seat shows up for the flight with probability \(p = 0.9\). Airlines often “ over book” flights. For an airplane with 300 seats, Bazinga airline accepts 325 reservations. Assume that passengers show up to the plane with probability \(p\) independent of each other. What is the chance that the flight is overbooked namely more than (\(> 300\)) passengers show up for the flight?

Use the normal approximantion with continuity correction.


6. Robin is teaching a large Statistics class. After computing the total grade of every student of his class (out of 100) he sees that if \(X\) denotes the score of a randomly selected student in the class then the distribution of this random variable has a Normal distribution with mean \(\mu = 80\) and \(\sigma = 6\). Robin decides his grading scheme will be as follows: Let \(G\) be the grade of the randomly selected student and note that \(G\) takes the values \(A, B, C, D\) and \(F\) . Fill in the following table:



7. Sherlock is hard on the trail of case. Watson was helping him do some calculations for the case but then got married and left for their honeymoon. Watson left some partial calculations for Sherlock but his handwriting is not great. The last note Watson left Sherlock was the following:

Sherlock knows that the expectation of the random variable \(X\) is \(\mu=5\). He really needs the variance \(\sigma^2\) of \(X\). Find this value.


8. If 54% of voters favor Candidate A over Candidate B, what is the probability that a simple random sample of 200 voters, a majority (\(\geq 101\)) favor Candidate B?


9. A student takes a multiple choice test where each question has 4 possible answers with only one correct and 3 others wrong. She gets the right answer if she knows it, otherwise she guesses at random. Assume the probability for her to know the answer to a question is \(0.8\). Suppose there are 100 questions on the test. Let \(X\) be the number of correctly answered questions. Assuming \(X\) has a Binomial distribution, find \[P(75 \leq X \leq 87).\] Use the normal approximation with continuity correction