Maths - Three Probability Problems

1. A salesman can contact either one or two customers per day with probability 1/3 and 2/3 respectively. Each contact will result in either no sale or a $20,000 sale with probability 9/10 and 1/10 respectively. What is the distribution of the random variable.

X = daily sales in $.

Use this distrbution to find the expected value and variance of his daily sales

2. The Parteo distribution,with paramters ,has pdf

a) Verify that f(x) is a pdf.

b) Derive the mean and variance of this distribution

c) Show that the variance does not exist, i.e, its value is infinetely large, if

3. Consider an exponential distribution with parameter B>0, i.e, its pdf is

a) Prove that its momento generating function, i.e,

b) Derive the pdf of the transformed random variable