If a random variable X is defined such that EX 12 10 and E

If a random variable X is defined such that E[(X 1)2 ] = 10 and E[(X 2)2 ] = 6. Find E(X) and V (X) .{please explain step by step}

Solution

As

E[(X - 1)^2] = E(X^2 - 2x + `1) = E(X^2) - 2E(X) + 1 = 10

Then

E(X^2) - 2E(X) = 9 [1]

Also,

E[(X - 2)^2] = E(X^2 - 4x + `4) = E(X^2) - 4E(X) + 4 = 6

Then

E(X^2) - 4E(X) = 2 [2]

Subtracting Equations [2] and [1],

2E(x) = 7

Thus,

E(x) = 3.5 [ANSWER]

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Thus,

E(X^2) = 9 + 2E(X) = 16

Thus,

V(X) = E(X^2) - E(X)^2 = 16 - 3.5^2 = 3.75 [ANSWER]