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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
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Solution
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`
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