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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
Solution
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
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