Question

If a continuous random process has a probability density function f(x) = a + bx, for 0 < x < 5, where a and b are constants, and P(X > 3) = 0.3 Determine:

1. The values of a and b.
2. The cumulative distribution function F(x)
3. P(X < 2)
4. P(2 < X < 4)

Answer 1

(a)

Given probability density function :

,

                          for 0 < x < 5

and

P(x > 3)= 0.3

To find a and b:

Since Total Probability = 1, we get:

                  (1)

We have:

So,

                       (2)

Substituting (2) equation (1) becomes:

Thus, we get:

                                        (3)

GiveP(x > 3)= 0.3

So, we get:

                 (4)

We have:

So,

                                (5)

Substituting (5), equation(4) becomes:

                                                   (6)

Thus, we have:
To solve:

                                          (3)

                                          (4)

5 X (4) gives:

                                     (5)

(5) - (3) gives:

15 a = - 0.5

So,

a = - 0.0333

Substituting in (3), we get:

- 0.8333 + 10 b = 2

So,

b = 0.2833

Thus, we get the required probability density function:

f(x) = - 0.0333 x + 0.2833,

                                    0 < x < 5

So,

Answer is:

a = - 0.0333

b = 0.2833

(b)

Cumulative Distribution Function F(x) is got as follows:

Integrating, RHS, we get Cumulative Distribution Function F(x) is got as follows:

, for 0< x < 5

           = 1 for x >5

          =0, otherwise

(c)

   between limits 0 to 2.

Applying limits, we get:

P(X<2)= 0.5

So,

Answer is:

0.5

(d)

between limits 2 to 4

Applying limits, we get:

P(2<X<4)= 0.3668

So,

Answer is:

0.3668