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Question
Find mean and standard deviation of the continuous random variable X whose p.d.f. is given by f(x) = 6x(1 - x);= (0); 0 < x < 1(otherwise)
Solution
Given p.d.f is
f(x) = 6x(l - x), 0 < x< 1
= 0 otherwise
Mean = E(X)
= `∫_0^1 x f(x)` dx
= `∫_0^1 x (6x - 6x^2)` dx
= `∫_0^1 6x^2 dx - 6 ∫_0^1 x^3 dx`
= `6[x^3/3]_0^1 - 6[x^4/4]_0^1`
= `6[1/3] - 6[1/4]`
∴ Mean = `2 - 3/2 = 0.5` ....(i)
`"Var" ("X") = "E"("X"^2) - ["E"("X")]^2`
= `∫_0^1 x^2 f(x) dx - (0.5)^2` .....[by (i)]
= `∫_0^1 x^2(6x - 6x^2) dx - (0.5)^2`
= `6∫_0^1 x^3 dx - 6 ∫_0^1 x^4 dx - 0.25`
= `6[x^4/4]_0^1 - 6[x^5/5]_0^1 - 0.25`
= `6[1/4] - 6[1/5] - 0.25`
= 0.3 - 0.25
Var.(X) = 0.05
S.D. (X) = `sqrt("Var".("X"))`
= `sqrt(0.05)`
S.D.(X) = 0.2236
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