Question

1. (6 pts) Consider a non-negative, discrete random variable X with codomain {0, 1, 2, 3, 4, 5, 6} and the following incomplete cumulative distribution function (c.d.f.): 0 0.1 1 0.2 2 ? 3 0.2 4 0.5 5 0.7 6 ? F(x) (a) Find the two missing values in the above table. (b) Let Y = (X2 + X)/2 be a new random variable defined in terms of X. Is Y a discrete or continuous random variable? Provide the probability mass function (p.m.f.) of Y. (c) Compute P(1.5 <Y < 10.5) 2. (7 pts) An urn consists of six distinct balls numbered 1 through 6. Suppose three balls are randomly chosen without replacement from the urn in a single trial. Let X be the highest number drawn among the three balls. What are the codomain and p.m.f. of X? 3. (7 pts) Consider that you are playing the following game: You toss a fair coin and earn $1 if the outcome is heads (H), and lose $1 if the outcome is tails (T). Initially you have $10 on hand, and you play this game until either you lose all your money or you have $20 on hand. (a) Let X be the number of tosses until the game ends. What is the codomain of X? Is this random variable discrete, continuous or neither? (b) Let Y be the amount of money you have on hand after three tosses of the coin. What is the codomain of Y? What is the probability distribution (p.m.f.) of Y?

Answer 1

Solution X F(x) 0 0.1 3 P 0.2 4 0.5 5 0.7 6 q 0.2 F i & cumulative Sum P(x=x) = f(_) - F(x) P(x-o) - FCO) - F(-) = 0.1-0 P(2:1) = f(1) - F(-) -0.2 -0.1 =0:1 P(x=2) = F(2) - F(20) 3 P-0.2 P(x = 3) = F(3)-F(30) - 0.2 -P p(:4), 0.5-0.2 = 0.3 p(x+5) = 0.7 -0.5 = 0.2 P (x=6) - Z-0.7 - 9-0.7 Scanned with CamScanner

{ Pi =1 6 IW N - OX O 3 0.2 4 0.5 5 0.7 6 1 0.1 0.2 0.2 let y = x² + x Yax? + X 0 - O M J Scanned with CamScanner

» Hence y take value discrete. - Y is a discrete random Variable. with PME Lol PO) 0.11 2 | | 6 10.3 | 3 | 4 | 5 0.3 0.9 | 15 21 لما | PGCE) 0.1 0.1 010 0.3 0.2 0.3 P (1.5 21 210.5) -PCY= 2) + P CY= 3) +---+ P CY=10) But y can take value y = 0,1,3,6,10,15,21 > p (Y=3) + P(4=6) + P (y=10) = 0 + 0 +0.3 = 0.3 Thank you Scanned with CamScanner