Question

A person tosses a coin and is to receive ₹ 4 for a head and is to pay ₹ 2 for a tail. Find the expectation and variance of his gains

Answer 1

Let X denote the amount the person receives in a game

Then X takes values 4, – 2 and

So P(X = 4) = P  ......(of getting a head)

= `1/2`

P(X = – 2) = P (of getting a tail)

= `1/2`

Hence the Probability distribution is

X	4	– 2
P(X = x)	`1/2`	`1/2`

E(X) = `sumx"P"_x (x)`

= `(4 xx 1/2) + (-2 x 1/2)`

= `2 + (- 1)`

E(X) = 1

E(x²) = `sum_x x^2"P"_x (x)`

= `[(4)^2 xx 1/2] + [(-2)^2 xx 1/2]`

= `[16 xx 1/2] + [4 xx 1/2]`

= 8 + 2

= 10

E(x²) = 10

Var(x) = E(x²) – E(x²)

= 10 – (1)²

Var(x) = 9

∴ His expected gain = ₹ 1

His variance of gain = ₹ 9