Question

The probability density function of a continuous random variable X is given by f(x) = x/2 for 0 < x < 2. Find its mean and variance.

Answer 1

The probability density function of a continuous random variable X is given by f(x) = x/2 , 0 < x < 2

We have to find mean .
Actually expected value of X is not other than mean .
Means, E(x) = mean,
And we know the relation between E(x) and f(x) when [ a, b] is Domain of f(x)
E.g., E(x) =ₐ∫ᵇxf(x) dx = ₀∫² x(x/2)dx
= [x³/6]₀²
= 8/6 = 4/3

Now, variable , Var(X) = E(X²) - E(X)²
E(X²) = ₀∫² x²(x/2) dx
= [x⁴/8]₀²
= 2

Now, Var(X) = 2 - (4/3)² = 2 - 16/9 = 2/9