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You take an exam that contains 20 multiple-choice questions. Each question has 4 possible options. You know the answer to 10 questions, but you have no idea about the other 10 questions so you choose answers randomly. Your score X on the exam is the total number of correct answers. Find the p.m.f. of X. What is P(X > 15)?
I roll a fair die repeatedly until a number larger than 4 is observed. If N is the total number of times that I roll the die, find P(N = k), for k = 1,2,3,....
Find the variance of the sampling distribution of means 2,5,7,8.
Each of the 50 students in class belongs to exactly one of the four groups A, B, C, or D. The membership numbers for the four groups are as follows: A: 5, B: 10, C: 15, D: 20. First, choose one of the 50 students at random and let X be the size of that student’s group. Next, choose one of the four groups at random and let Y be its size. (Recall: all random choices are with equal probability, unless otherwise specified.)
(a) Write down the probability mass functions for X and Y . (b) Compute E[X] and E[Y ] . (c) Compute V (X) and V (Y ). (d) Assume you have s students divided into n groups with membership numbers s1,s2,··· ,sn and again X is the size of the group of a randomly chosen student, while Y is the size of the randomly chosen group. Let E[Y ] = µ and V (Y ) = σ2. Express E[X] with s,n,µ, and σ.
You are dealt one card at random from a full deck and your opponent is dealt 2 cards (without any replacement). If you get an Ace, he pays you $10, if you get a King, he pays you $5 (regardless of his cards). If you have neither an Ace nor a King, but your card is red and your opponent has no red cards, he pays you $1. In all other cases you pay him $1. Determine your expected earnings. Are they positive?
Roll a fair die repeatedly. Let X be the number of 6’s in the first 10 rolls and let Y the number of rolls needed to obtain a 3.
(a) Write down the probability mass function of X.
(b) Write down the probability mass function of Y .
(c) Find an expression for P(X ≥ 6).
(d) Find an expression for P(Y > 10).
Given the Standard Normal distribution, find the following
(a) P(Z < 1.8)
(b) P(−1.1 < Z ≤ 1.8)
(c) P(−1.8 ≤ Z ≤−1.1)
(d) P(Z > −2.5)
(e) P(Z > −0.95)
(f) P(Z < −0.95)
(g) P(Z ≥ 2.18)
(h) P(Z > 10)
The number of claims per month paid by an insurance company is modelled by a random variable N with p.m.f satisfying the relation
p(n + 1) =
1/3p(n), n = 0,1,2,...
where p(n) is the probability that n claims are filed during a given month
(a) Find p(0).
(b) Calculate the probability of at least one claim during a particular month given that there have been at most four claims during the month.
The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem?
(a) The random variable Y has a Poisson distribution and is such that P(Y = 0) = P(Y = 1). What is P(Y 2 = 1)?
(b) Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?
