Question

Suppose we have the following pdf for the random variable X

f(x) ={x 0<=x<=1

c/x^2 1<=x<= infinity

0 otherwise

}

(a) 2 points Find the value c such that f(x) is a valid pdf.

(b) 3 points Find the cdf of X.

(c) 1 point Find the 75th percentile of X.

Answer 1

(a)

c is found by noting that the Total Probability = 1.

Thus, we get:

i.e.,

between limits1 to

Applying limits, we get:

0-(c) = 1

So,

c = 1

(b)

Thus,

pdf of X is given by:

,

                        for 1 X

The cdf of X is got by integrating f(x) between 1 to x as follows:

between limits 1 to x.

Applying limits, we get:

Thus, cdf of X is given by:

F(x) = 0 for x < 1

   for

(c)

75th percentile is got as follows:

i.e.,

between limits 1 to x..

Applying limits, we get:

So,

we get:

So

75th percentile = 4