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)

Answer 1

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