I need the answers ASAP please.
26. Directly from the definitions of expected value and variance, compute E(X ) and Var(X) when X has probability mass function given by the following table: X -2 -1 0 1 2 p(X) 1/15 2/15 3/15 4/15 5/15 27. Suppose that X takes values between 0 and 1 and has probability density function 2r. Compute Var(X) and Var(X?). 28. The random variable X takes values -1, 0, 1 with probabilities 1/8, 2/8, 5/8 respec- tively.(a) Compute E(X). (b) Give the pmf of Y = X and use it to compute E(Y). (c) Instead, compute E(X') directly from an extended table. (d) Compute Var(X). 29. Suppose X is a random variable with E(X ) = 5 and Var(X) = 2. What is E(X?)? 30. Compute the expectation and variance of a Bernoulli(p) random variable. 31. Suppose 100 people all toss a hat into a box and then proceed to randomly pick out a hat. What is the expected number of people to get their own hat back. Hint: express the number of people who get their own hat as a sum of random variables whose expected value is easy to compute. 32. Suppose I play a gambling game where I either win or lose & dollars. Suppose further that the chance of winning is p = .5. I employ the following strategy to try to guarantee that I win some money. I bet $1; if I lose, I double my bet to $2. if I lose I double my bet again. I continue until I win. Eventually I'm sure to win a bet and net $1 (run through the first few rounds and you'll see why this is the net). If this really worked casinos would be out of business. Our goal in this problem is to understand the flaw in the strategy. (a) Let X be the amount of money bet on the last game (the one I win). X takes values 1, 2, 4, 8, .... Determine the probability mass function for X. That is, find p(2* ), where k is in {0, 1, 2, . . . }. (b) Compute E(X). (c) Use your answer in part (b) to explain why the stategy is a bad one.31". Suppose you roll a fair Iii-sided die 10!] tima [independently]1 and you get $3 every time you roll a B. Let X1 be the number of dollars you win on rolls 1 through 25. Let X2 be the number of dollars you win on rolls 26 through 5|]. Let X; be the number of dollars you win on rolls 5] through i'ii. Let X4 he the number of dollars you win on rolls T6 throught ill]. Let X = X1+ X3 +333 + I, be the total number ofdoliars you win over all l rolls. {a} What is the probability mass function of X? (b) What is the expectation and variance of X? {c} Let F = nil}. [So instead of rolling ll times, you just roll 25 times and multiply your winnings by 4.} {i} 1What are the expectation and variance of 1"? [ii] How do the expectation and variance of 1"" compare to those of X? {I.e., are they bigger, smaller1 or equal?) Explain {briey} why this makes sense. 33. Let R be the rate at which customers are served in a queue. Suppose that R is exponential with pdf r} = Ee\" on [I], co}. Find the pdf of the waiting time per customer T = HR. 39. A continuous random variable X has PDF x] = x + org on [[1,1] Find a, the GDF and P[.5 s: X 4:: 1}. 4i]. {PL-[F of a sum] Suppose X and Y are independent and I m Bernoullij'} and Y m Bernouiiij). Determine the prof of X + Y 41. Let X be a discrete random variable with pmt'p given by: x 2 l [I 1 2 so] ins ans 3:15 4515 5,!15 {a} Let Y = XE. Find the pmfof'ir'. {b} Find the value the cdfof X at -1f2, 3H, TIE? 1? 1.51 5. {c} Find the value the cdf of?" at -1,:"2,. HILL, U3. 1, 1.5, 5
