Problem

Xhiver <xhiver@gmail.com> asks:

Let X~U[-1,2] and Y~exp(1)

(b) Let V = min {X,Y}. Find the CDF and PDF of V
(c) Let W = max {X,Y}. Find the CDF and PDF of W

Solution

Given:  we have a Uniform(-1,2) distribution with pdf f(x):

In[2]:=

… and an Exponential(1) distribution with pdf g(x):

In[3]:=

Mininimum of Uniform(-1,2) and Exponential(1)

This is just the special case of finding the distribution of the sample minimum, in a sample of size 2, where one random variable (f) has an Uniform(-1,2) distribution, and the other has an Exponential(1). The pdf of the minimum is simply:

In[4]:=

Out[4]=

... defined on the real line:

In[5]:=

Quick Monte Carlo check ...
Here are 100000 pseudo-random drawings from both the Uniform(-1,2) and the Exponential(1) distributions:

In[6]:=

Next, we break up the sample data into groups of size 2, and find the sample minimum of each pair … yielding our sampleMin data … which we then compare with the theoretical distribution hMin derived above:

In[7]:=

Out[8]=

Looks gooooood!

Find the cdf:

In[9]:=

Out[9]=

In[10]:=

Out[10]=

Max of Uniform(-1,2) and Exponential(1)

Here is the pdf of the maximum of a Uniform(-1,2) and an Exponential(1) random variable:

In[11]:=

Out[11]=

again defined on the real line:

Quick Monte Carlo check ...
Here again are 10000 pseudo-random drawings from the Uniform(-1,2) and from the Expontential(1) distributions:

In[13]:=

and here we break up the sample data into groups of size 2, and find the sample maximum of each pair … thus generating our sampleMax data … which we then compare with the theoretical distribution hMax derived above:

In[14]:=

Out[15]=

Find the cdf:

In[16]:=

Out[16]=

In[17]:=

Out[17]=

Created with Mathematica and mathStatica