Cdf technique

We can define the relationship of distributions and the property of distribution by using Cumulative distribution function, which is easily called cdf, F(X=x). 

In probability, Cumulative distribution function is the probability that X will take a less or equal value to a real value x, which can be expressed as P(X<=x).

We will use the concept that the derivative of Cumulative distribution function in terms of random variable becomes the Probability density function of the random variable. 

So we can find pdf of some distributions by differentiating the cdf. 

There are some examples that I found in some textbooks. 

<Examples for finding distributions by using CDF Technique>

1.

When the distribution of X is given, we can find the distribution of Y which is the function of X by using PDF of X. 

Firstly, as our objective is to find the pdf of Y. For using CDF Technique, firstly we set the CDF of Y and then, substitute the random variable Y with the function of X which is -λlogX. Then we make the Inequality in the probability in terms of the random variable X. By using the known CDF of X, we can find the CDF of Y and we finally differentiating it and able to find the pdf of X. 

2.

3.

You should take care of the range of random variable X and Y here. Even though you can easily know the range of Z should be (0,2), It cannot be possible to make Z be 2 by setting X be 1.5 and Y be 0.5 because the range of X should be between 0 and 1. So If you are given with a distribution which have the range, You should take care of not exceeding the given ranges. So you must take care of two cases which are 0<=Z<=1 and 1<=Z<=2 separately. 

<Proof for additivity of Poisson distribution - CDF Technique>

<Proof for relation between Standard normal distribution and Chi-square distribution with 1 degree of freedom - CDF Technique>