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