Chapter 3, Problem 7CE

marks 11 3 3/4 x 1/4 1. There are 4 balls in an urn, of which 3 balls are white and 1 ball is black. You do the following: draw a ball from the urn at random, note its colour, do not return the ball to the urn; draw a second ball, note its colour, return the ball to the urn; finally draw a third ball and note its colour. (i) Describe the corresponding discrete probability space (Q, F, P). [9 Marks] (ii) Consider the following event, A: Among the first and the third balls, one ball is white, the other is black. Write down A as a subset of the sample space and find its probability, P(A). [2 Marks]

There are 4 balls in an urn, of which 3 balls are white and 1 ball isblack. You do the following:• draw a ball from the urn at random, note its colour, do not return theball to the urn;• draw a second ball, note its colour, return the ball to the urn;• finally draw a third ball and note its colour.(i) Describe the corresponding discrete probability space(Ω, F, P). [9 Marks](ii) Consider the following event,A: Among the first and the third balls, one ball is white, the other is black.Write down A as a subset of the sample space Ω and find its probability, P(A)

Let (Ω, F, P) be a probability space and let X : Ω → R be a randomvariable whose probability density function is given by f(x) = 12 |x|e−|x| forx ∈ R.(i) Find the characteristic function of the random variable X.[8 Marks](ii) Using the result of (i), calculate the first two moments of therandom variable X, i.e., E(Xn) for n = 1, 2. [6 Marks]Total marks 16 (iii) What is the variance of X?

Let X be a random variable with the standard normal distribution, i.e.,X has the probability density functionfX(x) = 1/√2π e^-(x^2/2)2 .Consider the random variablesXn = 20(3 + X6) ^1/2n e ^x^2/n+19 , x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limit exists and find it limn→∞E(Xn)

Let X be a discrete random variable taking values in {0, 1, 2, . . . }with the probability generating function G(s) = E(sX). Prove thatVar(X) = G′′(1) + G′(1) − [G′(1)]2.[5 Marks](ii) Let X be a random variable taking values in [0,∞) with proba-bility density functionfX(u) = (5/4(1 − u^4, 0 ≤ u ≤ 1,0, otherwise.     Let y =x^1/2 find the probability density function of Y

14 14 4. The graph shows the printing rate of Printer A. Printer B can print at a rate of 25 pages per minute. How does the printing rate for Printer B compare to the printing rate for Printer A? The printing rate for Printer B is than the rate for Printer A because the rate of 25 pages per minute is than the rate of for Printer A. pages per minute RIJOUT 40 fy Printer Rat Number of Pages 8N WA 10 30 20 Printer A 0 0 246 Time (min) X

2. y 1 Ο 2 3 4 -1 Graph of f x+ The graph gives one cycle of a periodic function f in the xy-plane. Which of the following describes the behavior of f on the interval 39 x < 41 ? (Α B The function f is decreasing. The function f is increasing. The function f is decreasing, then increasing. D The function f is increasing, then decreasing.

Depth (feet) 5- 4- 3- 2. WW www 1 D B 0 10 20 30 40 50 60 70 80 Time (hours) x A graph of the depth of water at a pier in the ocean is given, along with five labeled points A, B, C, D, and E in the xy-plane. For the time periods near these data points, a periodic relationship between depth of water, in feet, and time, in hours, can be modeled using one cycle of the periodic relationship. Based on the graph, which of the following is true? B C The time interval between points A and B gives the period. The time interval between points A and C gives the period. The time interval between points A and D gives the period. The time interval between points A and E gives the period.

A certain type of machine produces a number of amps of electricity that follows a cyclic, periodically increasing and decreasing pattern. The machine produces a maximum of 7 amps at certain times and a minimum of 2 amps at other times. It takes about 5 minutes for one cycle from 7 amps to the next 7 amps to occur. Which of the following graphs models amps as a function of time, in minutes, for this machine? A B C D Amps M 3 4 5 678 Minutes Amps w 3 4 5 6 7 8 Minutes 8 Amps- 6+ Amps y 2345678 Minutes 456 8 Minutes