A random variable X has a cumulative distribution fnc given by
0 for a<0
G(a)={a^2 for 0<=a<1
1 for a>=1
find the probability density function.
To find the probability density function (pdf), we need to take the derivative of the given cumulative distribution function (cdf) with respect to the random variable X.
Let's differentiate the cdf in each interval:
For a < 0:
Since the cdf is constant before 0, the derivative is 0.
For 0 <= a < 1:
Differentiating a^2 with respect to a, we get 2a.
For a >= 1:
Since the cdf is constant after 1, the derivative is 0.
So the pdf will be:
0, for a < 0
2a, for 0 <= a < 1
0, for a >= 1
Therefore, the probability density function (pdf) of the random variable X is as follows:
f(a) = 0, for a < 0
f(a) = 2a, for 0 <= a < 1
f(a) = 0, for a >= 1