Question

Q–2: [5+2+3 Marks] Let X be a random variable giving the number of
heads minus the number of tails in three tosses of a coin.
a) Find the probability distribution function of the random variable X.
b) Find P(−1 ≤ X ≤ 3).
c) Find E(X).

Answer 1

Formulas: 1) If P(X-x) is probability distribution function of the discrete random variable X then E(X) xx * P(X-x). Given that X is a random variable gives the number of heads minus the number of tails in 3 tosses of a coin. Let H number of heads in 3 tosses of a coin T number of tails in 3 tosses of a coin. In three tosses of a coin the number of possible outcomes are 8. In 3 tosses of a coin H 0, 1,2,3

Number of possible ways to get 0 heads and 3 tails [TTT] 1 P(X =-3) = P(0 heads and 3 tails) = 8 Number of possible ways to get 1 head and 2 tails [(HTT), (THT). (TTH)] 3 P(x1P(l head and 2 tails)-g Number of possible ways to get 2 heads and 1 tai[(HHT), (HTH), (THH)] 3 P(X- 1)- P(2 heads and 1 tail)- Number of possible ways to get 3 heads and 0 tails [HHH 1 PC-3)-P(3 heads and 0 tails) = 8

a) Now we have to find the probability distribution function of the random variable X- P(X -x) -1 P(X = x) 8 b) Now we have to find P(-1s X s 3). PC-1 X 3) = P(X =-1) + P(x = 1) + P(X = 3)

c) Now we have to find E(X) (-3 * P(X =-з)) + (-1 * P( -1)) + (1 * P(X = 1)) + (3 * P(X = 3)) 0