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Question
Consider a random variable X with p.d.f.
f(x) = `{(3x^2",", "if" 0 < x < 1),(0",", "otherwise"):}`
Find E(X) and V(3X – 2)
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Solution
Let X be the random variable
`"E"(x^2) = int_(-oo)^oo x"f"(x) "d"x`
`"E"(x) = int_0^1 x(3x^2) "d"x`
= `int_0^1 x(3x^3) "d"x`
= `3[x^4/4]_0^1`
= `3/4[x^4]_0^1`
= `3/4[1 - 0]`
`"E"(x) = 3/4`
`"E"(x^2) = int_(-oo)^oo x^2"f"(x) "d"x`
= `int_0^1 x^2 (3x^2) "d"x`
= `int_0^1 3x^4 "d"x`
= `3(x^5/5)_0^1`
= 3/5[x^5]_0^1`
= `3/5[1 - 0]`
= `3/5`
Var(x) = `"E"(x^2) - ["E"(x)]^2`
= `33/5 - (3/4)^2`
= `3/5 - 9/16`
= `(48 - 45)/80`
Var(x) = `3/80`
`"v"(3x - 2) = (3)^2"Var"(x)` .......`{because "v"(""x + "b") = "a"^2"v"(x)}`
= `9(3/80)`
∴ `"V"(3x - 2) = 27/80`
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