Question

The probability density function of a continuous random variable X is
f(x) = `{{:(a + bx^2",",  0 ≤ x ≤ 1),(0",",  "otherwise"):}`
where a and b are some constants. Find Var(X)

Answer 1

`"E"(x^2) = int_(-oo)^oo x^2"f"(x)  "d"x`

= `int_0^1 ("a"x^2 + "b"x4) "d"x`

= `{3/5 (x^3/3) + 6/5[x^5/5]}_0^1`

= `[1/5 (x^3) + 6/25 (x^5)]_0^1`

= `[1/5 (1) + 6/25(1)] - [0]`

= `1/5 + 6/25`

= `(5 + 6)/25`

= 11

Var(x)= `"E"(x^2) - ["E"(x)]`

= `1/25 - (3/5)^2`

= `11/25 - 9/5`

= `(11 - 9)/25`

Var(x) = `2/25`