Probability Theory and Statistics

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Probability Theory and Statistics

ENGR 3341 Probability Theory and Statistics Prof. Gelb Homework #3

Suggested “warm-up” problems from textbook (answers on eLearning):

Section 3.1.6: problems 1, 3, 6 Section 3.3: problems 1, 5, 10, 11

Problems to be turned in:

1. 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

  1. 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”.
    1. (a) What’s the probability of choosing three Skittles and getting three of the same flavor?
    2. (b) What’s the probability of choosing five Skittles and getting one of each flavor?
    3. (c) What’s the probability that in a bag of 30 candies there aren’t any Lemon ones?
    4. (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?
  2. 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.
  3. Suppose X ∼Geometric(0.3). (a) Calculate P(X ≤ 3).
    (b) Calculate E[X]
    (c) Calculate P(X ≥ 4|X ≥ 2)
  4. SupposeY ∼Binomial(6,0.5) (a) Calculate P(Y ≥ 3). (b) Calculate μY
    (c) Calculate P(Y ≥ μY )
  5. 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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