Question

Please give me an example with solution for each:

a) Know how to take probabilities involving the exponential, gamma, and normal distributions (for the gamma you will either be asked to just leave it as an integral or will be allowed to use technology.)

Answer 1

to compute the probability you need to know the probability density function(continuous random variable), probability mass function (discrete random variable).

Exponential distribution:  

is defined by Exp()

PDF(probability distribution function) is given by

so once the parameter is known we can compute the probability for each value of x

Gamma distribution:

is defined by

where are known for the distribution

The corresponding probability density function is

where is the gamma function. So for any given x you can compute the probability

The cumulative distribution function is:

so you have to integrate to get the probability between the range

Normal Distribution:

is defined by

where is the mean of the distribution and is the standard deviation

The probability density of the normal distribution is