Please help me to answer these questions for cumulative distribution of the random variable X

Question

The cumulative distribution of the random variable X is given by

G(x) = 0                    for x<0
           3x²-2x³          for 0≤x<1
           1                   for x ≥1

a)Find the probability density function g(x).
b)Use the cumulative distribution G(x)to compute P(X < ¼) and P(X> ½)
c)Find the mean of the random variable x.
d)Find the standard deviation of the random variable

Answer 1

The density function g(x) = G'(x) = 6x - 6x² between ) and 1 and 0 elsewhere.

 

P(X<1/4) = G(1/4) and P(X>1/2) = 1 - P(X<1/2) = 1/2

 

μ = ∫6x²  - 6x³ dx between 0 and 1 = 1/2

 

σ² = ∫(x-μ)² g(x) dx = ∫x² dx - μ² between 0 and 1...and I will leave it to you to compute this last integral.