1 Suppose X is a random variable vit1i density function a Fi

1: Suppose X is a random variable vit1i density function a) Find P(X leq 1/2). b) Find P(X geq 3/4). c) Find P(X geq 2). d) Find E[X]. e) Find the standard deviation of X.

Solution

a) P(X<= 1/2 )

= integral (from 0 to 1/2) 2xdx

= integral (from 0 to 1/2) ( x^2)

= 1/4 Answer

b) P(X>= 3/4 )

= integral (from 3/4 to 1) 2xdx

= integral (from 3/4 to 1) ( x^2)

= (1 - 9/16)

= 7/16 Answer

c) P(X>= 2 )

= integral (from 2 to infinity) 0 dx

= 0 Answer

d) E(X)

= integral (from 0 to 1) (x* 2x)dx

= integral (from 0 to 1) ( 2x^2)dx

= integral (from 0 to 1) ( 2x^3/3)

= 2/3 Answer

e)

E(X^2)

= integral (from 0 to 1) (x^2* 2x)dx

= integral (from 0 to 1) ( 2x^3)dx

= integral (from 0 to 1) ( x^4/2)

= 1/2

Variance = E(X^2) - E^2(X) = 1/2 - 4/9 = 1/18

Therefore Standard Deviation = sqrt(Variance) = sqrt(1/18) = 0.235 Answer