Please Help me to solve this

Question

The density function of X is given by f (x) =2x, 0≤x ≤1 0, otherwise Find variance of X.

Answer 1

Hi Shuvo!

The variance of X is equal to the second moment of X, E(X²) minus the square of its mean, E(x).

Symbolically, this is expressed as σ² = E(X²)-μ^2, where μ = E(X).

So let's first find μ=E(X).

For your density function, E(X) is the integral from 0 to 1 of xf(x), which is 2x². The integral of 2x² is (2/3)x³, and when we evaluate that between 0 and 1, we find that μ=E(X)=2/3. So μ² = 4/9.

The second moment of X, E(X²) is the integral from 0 to 1 of x²f(x), wich is 2x³. The integral for 2x³ is (1/2)x⁴, and when we evaluate that between 0 and 1, we find that E(X²)=1/2.

So σ² = 1/2-4/9 = 9/18 - 8/18 = 1/18.

Hope this makes sense. Let me know if you need any other help.

Thanks,

Bob