True or False (1, 2)

1. If a continuous random variableX has zero mean, then its density f(x) has to be strictly positive at zero.

2. If f(x) is a density function on [0,1], then integral 0 to 1 f^2 (x) dx < infinity.

3. A density function is verbally described as follows: it is zero for x < 1, rises linearly between 1 and 2 to 1/3, remains constant between 2 and 4, decreases linearly to zero from 4 to 5, and remains zero thereafter.

a) Plot the density function

b) Find the corresponding CDF and plot it

c) Find the mean of the distribution

d) Find P(2.5 < X < 4.5)

** Please see the attached file for the complete solution response **

Problem #1

Answer: False

We consider the density function (please see the attached file) with (please see the attached file), then (please see the attached file) and …

This solution provides a detailed statistics tutorial showing how to plot the given density function, find the corresponding CDF and plotting it, and find the mean of the distribution. It includes completed graphic illustrations.