The random variable Y has PMF:
(a) Determine c such that this PMF is normalized.
(b) What is P(Y ≤ 1)?
(c) WhatistheexpectedvalueofY?
c(1/2)y
PY(y)= 0
y=0,1,2,3
otherwise
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Bags of “Original Skittles” candies come in five different flavors: Grape, Lemon, Green Apple, Orange
and Strawberry. The flavors are equiprobable; if you take a candy randomly from a bag the probability of
getting a particular flavor is 1/5, independent of other “draws”.- (a) What’s the probability of choosing three Skittles and getting three of the same flavor?
- (b) What’s the probability of choosing five Skittles and getting one of each flavor?
- (c) What’s the probability that in a bag of 30 candies there aren’t any Lemon ones?
- (d) if you draw a sequence of candies from the bag, what’s the probability that you’ll get at least two
Lemon ones before you get your first Grape one?
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Two teams A and B play a best-of-three games series; the series ends as soon as one team has won two
times. Assume the teams are equally matched, so the probability of A winning is 0.5 in every game.
There are no draws. Find:
(a) The PMF of N, the total number of games played.
(b) The expected value of N.
(c) The PMF of W , the number of times team A wins.
(d) The PMF of X, how many games the winner leads by at the end of the series. -
Suppose X ∼Geometric(0.3).
(a) Calculate P(X ≤ 3).
(b) Calculate E[X]
(c) Calculate P(X ≥ 4|X ≥ 2) -
SupposeY ∼Binomial(6,0.5)
(a) Calculate P(Y ≥ 3).
(b) Calculate μY
(c) Calculate P(Y ≥ μY ) -
Customers arrive at a bank randomly, but at an average rate of 1 per 10 minutes; the number that arrive in
t minutes is a Poisson random variable C with λ = 0.1t.
(a) What is the probability that exactly three customers will arrive in a given 5 minute interval?
(b) What is the probability that at least one customer arrives in a given 10 minute interval?
(c) What is the mean number of customers that will arrive between 8 AM and 9:30 AM?
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