Question

Write solutions legibly, and show all work. Walk the reader through your thought process, using English words when necessary.

1. Recall question 2 of the previous homework – We draw 6 cards from a 52 card deck and let X = the number of heart cards drawn. You already found the pmf back then. You’re allowed to use it here without re-deriving it. a. What is the expected value of X? b. What is the variance of X? What is the standard deviation of X?

2. A man has seven keys on a key ring, one of which fits the door he wants to unlock. He randomly selects a key and tries it. If it doesn’t work, he selects a different key at random and tries that one (so he’s sampling without replacement). He continues in this manner until the door opens. Let X = the number of keys he tries until opening the door (counting the key that actually works). Find the expected value and standard deviation of X.

3. Suppose that the average number of equipment breakdowns at a factory per week is 5. Later we’ll show that the average is our best guess at the expected value of ? = ?ℎ? ?????? ?? ????????? ?????????? ?? ? ????? ???? (assuming that this distribution stays the same across multiple weeks.) It’s also known that, through some other sample statistic, that our best guess at the standard deviation of X given the data is 0.8. a. With the help of our boy Chebyshev, find an interval such that we can be at least 90% that the number of breakdowns next week will fall within that range. b. Suppose that the supervisor promises the board of directors that the number of breakdowns will rarely exceed 8 in a one- week period. Is the supervisor safe in making this claim? Why or why not? c. Google Chebyshev and just stare at this guy’s magnificent beard for a few seconds. You don’t need to write anything down. You’re welcome.

4. Let ? be some random variable with mean ? and standard deviation ?. Define ? = ?−? ? . What is the mean of ?? What is the standard deviation of ?? (This general fact about random variables will come in handy later when we get to the normal distribution.)

5. Suppose we toss a fair coin twice. Let X = the number of heads, and Y = the number of tails. X and Y are clearly not independent. a. Show that X and Y are not independent. (Hint: Consider the events “X=2” and “Y=2”)
b. Show that E(XY) is not equal to E(X)E(Y). (You’ll need to derive the pmf for XY in order to calculate E(XY). Write down the sample space! Think about what the support of XY is and find those probabilities.)

6. Let ?~???????[0,?], let X = ???(?), and Y = ???(?) a. First, we’ll show that X and Y are not independent (which should be obvious but we still need to be able to show it.) i. What is ?(√2 2 ≤ Y ≤ 1)? (Hint: Put this probability in terms of U) ii. What is ?(√2 2 ≤ ? ≤ 1 | √2 2 ≤ X ≤ 1)? (Hint: ↑) b. What we’ve shown so far is that there is a clear causal relationship between X and Y. Next we’ll show that, despite X and Y being very, very dependent, they are still nonetheless uncorrelated. i. Find E(X), E(Y), and then multiply them together to get E(X)E(Y). (Hint: Use the Law of the Unconscious Statistician) ii. Find E(XY). (Hint: ↑) iii. The correlation between Y and Z is the square root of E(XY)-E(X)E(Y). What is it?
(If you’re interested in what this means in terms of the larger correlation vs causation argument, see the note at the end of the assignment)

7. Suppose that the mean exam score for a college class is a 60 (where the highest possible score is 100), and the standard deviation is 10. If there are 50 people in the class, what is the maximum possible number of A’s that there could have been? (Assume that getting an A means getting a score greater than or equal to 90) (Hint: What does Chebyshev’s Inequality tell you about scores between 30 and 90? That is, scores which are within 3 standard deviations of the mean?))

Answer 1

1:

Here X has hypergeometric distribution with following parameters:

Population size: M = 13

Number of successes in population, that is number of hearts: k=12

Sample size: n=6

(a)

The mean is

(b)

The variance is

The standard deviation is