Question

Let X be a continuous random variable with probability density function
f(x) = `{{:(3/x^4",",  x ≥ 1),(0",",  "otherwise"):}`
Find the mean and variance of X

Answer 1

Let x be a continuous random variable.

In the probability density function

Mean E(x) = `int_oo^oo x"f"(x)  "d"x`

Here E(x) = `int_1^oo x(3/x^4)`

= `3int_1^oo 1/x^3  "d"x`

Here E(x) = `int_1^oo x^-3  "d"x`

= `3[x^(-3 + 1)/(-3 + 1)]_1^oo`

= `3int_1^oo x^-3  "d"x`

= `3(x^(-3 + 1)/(-3 + 1))_1^oo`

= `3/(-2) (x^2)_1^oo`

= `(-3)/2 [1/x^2]_1^oo`

= `3/(-2) [1/oo^2 - 1/(1)^2]`

= `- 3/2 [0 - 1]`

= `3/2`

∴ E(x) = `3/2`

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

= `int_1^oo  x^2(3/x^4)  "d"x`

= `3/x^2  "d"x`

= `3int_1^oo x^2  "d"x`

= `3 [(x^-2 + 1)/(-2 + 1)]`

= `3[x^-1/(-1)]`

= `-3[1/x]_1^oo`

= `-3[1/oo - 1/1]`

= `-3[0 - 1]`

= 3

`"E"(x^2)` = 3

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

= `3 - (3/2)^2`

= `3 - 9/4`

= `(12 - 9)/4`

∴ Var(x) = `3/4`