Answer to Question #199472 in Statistics and Probability for sss

Question #199472

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


1
Expert's answer
2021-05-28T08:54:24-0400

A.

Function f(x) is a probability density function in range a to b if

abf(x)dx=1\int^b_af(x)dx=1

Then:

01(6x(1x))dx=(3x22x3)01=32=1\int^1_0(6x(1-x))dx=(3x^2-2x^3)|^1_0=3-2=1


C.

E(X)=abxf(x)dx=01x(6x(1x))dx=(2x31.5x4)01=21.5=0.5E(X)=\int^b_axf(x)dx=\int^1_0x(6x(1-x))dx=(2x^3-1.5x^4)|^1_0=2-1.5=0.5


B.

P(x<b)=0bf(x)dxP(x<b)=\int^b_0f(x)dx

P(x>b)=10bf(x)dxP(x>b)=1-\int^b_0f(x)dx


Then:

0bf(x)dx=10bf(x)dx\int^b_0f(x)dx=1-\int^b_0f(x)dx

20bf(x)dx=20b(6x(1x))dx=2(3x22x3)0b=6b24b3=12\int^b_0f(x)dx=2\int^b_0(6x(1-x))dx=2(3x^2-2x^3)|^b_0=6b^2-4b^3=1

b=0.5b=0.5



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