Question

For the probability density function.

f(x) = {Cx² ; 0≤ n ≤ 3
0 ; otherwise}

Find the value of C and P (1≤ n ≤ 2). Also find distribution function.

Answer 1

To solve this question we need to state the condition for formation of a probability distribution function.

CONDITIONS

1.) ₀∫°°f (x) dx = 1 0 ≤ x ≤ ∞

2.) f (x) ≥ 0 for all the values of x.

CALCULATIONS :

Taking the first condition and Applying it to the question :

₀∫³ Cx² = 1

C [ x³ / 3 ] ³₀ = 1

C [ 27 / 3] = 1

9C = 1

C = 1/9

The pdf is thus :

1/9x² Pr( 1 < x < 2) Integration of 1/9x^2 from 1 to 2. 1/9 [ 8/3 - 1 / 3] 1/9 × 7/3 = 7/27 Answer : 7 / 27