Question

Let X denote the amount of time for which a statistics reference book, on two-hour reserve at the library, is checked out by a randomly selected student. Suppose that X has the following probability density function (pdf) f(x)=kx2(2−x), 0≤x≤2.

(a) Find the cumulative distribution function (cdf) of X (Hint: Don’t forget that first you have to find k).

(b) Find the probability that the book is checked out for a time between 0.5 and 1.5 hours.

(c) Find the probability that the book is checked out for less than 1.5 hours, given that is checked out for more than 1 hour. 5

(d) Find the expected amount of time for which the book is checked out.

Answer 1

for this to be valid: f(x) dx must be 1

f(x) dx =k(2x²-x³) dx =k*(2x³/3-x⁴/4)|²₀ =k*(4/3)=1

k=3/4

a)

CDF =F(x)=P(X<x)= f(x) dx= (3/4)(2x²-x³) dx =(3/4)*(2x³/3-x⁴/4)|^x₀ =(3/4)*(2x³/3-x⁴/4)

b)

P(0.5<X<1.5)=F(1.5)-F(0.5)=(3/4)*(2*1.5³/3-1.5⁴/4)-(3/4)*(2*0.5³/3-0.5⁴/4)=0.6875

c)

P(X<1.5|X>1)=P(1<X<1.5)/P(X>1.5) =(F(1.5)-F(1))/(1-F(1.5))=(0.7383-0.3125)/(1-0.3125)=0.6193

d)

expeted time =E(X)=x f(x) dx =(3/4)(2x³-x⁴) dx =(3/4)*(2x⁴/4-x⁵/5)|²₀ =6/5 =1.2