Question

Verify whether the following are probability density function
(i) f (x) = 6x⁵, 0 < x < 1
(ii) f (x) = 2x/9, 0 < x < 3

Answer 1

Concepts :- conditions of probability density function :-
1. fₓ(x) ≥ 0 for all $\bold{X\in\mathbb{R}}$
2. $\bold{\int\limts_{-\infty}^{\infty}{f_X(u)}\,du=1}$
3. $\bold{P(a \ \textless \ X \leq b) = F_X(b)-F_X(a)=\int_{a}^{b} f_X(u)du}$

(i) f(x) = 6x⁵ , 0 < x < 1
Apply all conditions
1. f(x) ≥ 0 for all x ∈(0,1)
2. ∫6x⁵ dx = x⁶ = [ 1⁶ - 0⁶ ] = 1
Third one is similar case of 2nd .
Hence, this is probability density function.

(ii) f(x) = 2x/9 , 0 < x < 3
Apply all conditions
1. f(x) ≥ 0 for all x belongs to (0, 3)
2. ∫2x/9 dx = 1/9 [ x²] = 1/9 [ 3² - 0 ] = 1
3. 3rd condition is similar of 2nd
Hence, this function is also probability density function.