A) Show that f(x) is a p.d.f. B) Determine b. C) Find the mean of X.

Question

The diameter of an electric cable, say X, is assumed to be continuous random variable with p.d.f

1. f(x)=6x(1-x), 0 \< x \<1.

A. Show that f(x) is a p.d.f

B. Determine a number b such that P(X<b)=P(X>b)

C. Find the mean of X.

Answer 1

f(x) is a pdf if the area between f(x) and the x axis from x=0 to 1 is 1

this area is the integral from x = 0 to x = 1 of f(x) = 6x - 6x²

integrating gives 6x² / 2 - 6 x³/ 3 = 3x² - 2x³

when x = 1, this is 3-2 = 1

when x = 0 this is 0-0 = 0

so the integral value is 1-0=1 as required

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for (b) we choose b so that the integral from 0 to b = integral from b to 1

as the total area is 1, each area must be 0.5

ie 3b² - 2b³ - 0 = 0.5

3b² - 2b³ = 0.5

3b² - 2b³ - 0.5 = 0

you could plot this on Desmos (use x instead of b) and find the zero (0.5 works)

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To find the mean of X, integrate x f(x) from x = 0 to 1

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