I know this is a lot of questions, if you can answer any or all, I appreciate the help.
1. Suppose you keep on tossing a coin with P(head)=0.6 until you see the first head. Let X be the random variable that counts how many tosses you needed to see the first head. For this X=# tosses needed to get the 1st head, write the p.m.f table (first 5 or 6 entries). Then find the conditional probability that you will need more than 10 tosses to get the 1st head, given that the first 5 tosses haven't produced any heads. Finally, find the conditional probability that you will need more than 20 tosses to get the 1st head, given that the first 15 didn't produce any. Comparing these two conditional probabilities, what is your conclusion? [X is a geometric random variable]
2. Suppose you keep on tossing a coin with P(head)=0.6 until you see the 2nd head appearing (e.g. if you get HH, you stop right there at the 2nd toss; if you get THH or HTH, you stop at the third toss, etc). Once again, let X=# tosses needed to see the 2nd head. Write down the p.m.f. table for X [X is the negative binomial random variable].
3. Using factorial notation, write down a formula for the m-th number in the n-th row of Pascal's triangle. In other words, if I ask you to find the 15th number in the 50th row of Pascal's triangle, what formula will you use?
4. Suppose you go to a dairy farm that has a total of N cows, K of which are Holstein cows and (N - K) are Jersey cows. You randomly choose n cows out of N (where N, K and n are positive integers with n
5. Derive the results for a uniform [a,b] random variable (i.e. prove that the mean formula and variance formula are correct by computing them.
Quick Reply
