Question

Three coins are tossed simultaneously; X is the number of heads. Find the expected value and variance of X.

Answer 1

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} and X = {0, 1, 2, 3}

X = xi	P = pi	xipi	`bb(x_i^2p_i)`
0	`1/8`	0	0
1	`3/8`	`3/8`	`3/8`
2	`3/8`	`6/8`	`12/8`
3	`1/8`	`3/8`	`9/8`
		`sum_(i = 1)^n x_ip_i = 12/8`	`sum_(i = 1)^n x_i^2p_i = 24/8`

Then E(X) = `sum_(i = 1)^n x_ip_i = 12/8` = 1.5

Var(X) = `(sum_(i = 1)^n x_i^2p_i) - (sum_(i = 1)^n x_ip_i)^2`

= `24/8 - (1.5)^2`

= 3 − 2.25

= 0.75