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)

Answer 1

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`