Question

1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? (5) c. Compute P(A), P(B), P(A|B), and P(BA). [7] 2. Let U be a continuous random variable with the following probability density function: 1+t -1<tso g(t) 1- t 05t51 0 otherwise a. Verify that g(t) is indeed a probability density function. (5) b. Compute the expected value, E(U), and variance, V(U), of U. (10)

Answer 1

H A fair Coin is tossed 3 times a) Save Tree diagram H T H HHH HHI HTH HTT THE THT TTH ΤΤΤ. T H Start Н b) sample space S- SHHH, HTH HHT, THH, HTT, THT, TTH, ITT 13. nis)= 8 head Probability function P(x=r) = ne prann. P= getting" (one time rolled) = ² q=1-P=2 using binomial distribution, P(x=9 )= verp?q=*= 36, (+)* A = Platteast 2 heads) =P(x22) = P(x=2)+P(x-3) 2 3 2 (₂ 3 + 3 C3 (₂) = 3 + 2 = 4 / 2 = £ B=Plenactly 2 heads)= P(x-2)= 3c, () . PLAUB)= P(B)=3 PLAIB) = P(AB) PIB) AUB=B. -3/8 =1 3/8 = 3 P(BIA) = PIBOA) = PIA) 318 Tz 4