Question

Suppose that X is a continuous random variable with a probability density function that is a positive constant on the interval [8,20], and is 0 otherwise.

a. What is the positive constant mentioned above?

b. Calculate P(10?X?15).

c. Find an expression for the CDF FX(x). Calculate the following values.

FX(7)=

FX(11)=

FX(30)=

Answer 1

Given:

f(x) = a, in the interval 8 x20

     = 0, otherwise

The value of a is got by noting that the Total Probability = 1

i.e.,

i.e.,

a(20-8) = 1

So,

a = 1/12 = 0.0833

(b)

So, pdf of x is written as:

f(x) = 1/12, 8 x 20

    = 0, otherwise

(c)

(i)

CDF of X is got by integrating f(x) from 8 to X as follows:

(ii)

    since pdf f(x) is defined on in the interval [8,20].

(iii)

(iv)

F(30) = 1,

as f(x) is defined in the interval [8,20].