Obtain a raw moment for a continuous distribution:

Obtain the mean of a discrete distribution:

Obtain the variance of a truncated distribution:

An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder's loss,

, follows a distribution with density function

for

and 0 otherwise. Find the expected value of the benefit paid under the insurance policy:

An insurance company's monthly claims are modeled by a continuous, positive random variable

, whose probability density function is proportional to

where

. Determine the company's expected monthly claims:

Claim amounts for wind damage to insured homes are independent random variables with common density function

for

and 0 otherwise, where

is the amount of a claim in thousands. Suppose 3 such claims will be made. Compute the expected value of the largest of the three claims:

Let

represent the age of an insured automobile involved in an accident. Let

represent the length of time the owner has insured the automobile at the time of the accident.

and

have joint probability density function

for

and

, and 0 otherwise. Calculate the expected age of an insured automobile involved in an accident:

A baseball player is a 0.300 hitter. Find the expected number of hits if the player comes to bat 3 times:

A basketball player shoots free throws until he hits 4 of them. His probability of scoring in any one of them is 0.7. Find the number of shots the player is expected to shoot:

Four six-sided dice are rolled. Find the expectation of the minimum value:

Find the expectation of the maximum value:

Find the expectation of the sum of the three largest values. Using the identity

and linearity of

Expectation you get:

A random sample of size 10 from a continuous distribution

is sorted in ascending order. A new random variate is generated. Find the probability that the 11

sample falls between the fourth and fifth smallest values in the sorted list:

The probability equals

and is independent of

:

It is also independent of the distribution:

Study the tail value at risk (TVaR) for the exponential distribution:

Value at risk may underestimate possible losses. Consider two models for stock log-returns:

Fix parameter

so that values at risk at the 99.5% level are equal:

Now compute the expected losses in both models, given that they exceed the value at risk:

The losses are actually bigger in the second model:

A drug has proven to be effective in 40% of cases. Find the expected number of successes when applied to 100 cases:

Assuming stock logarithmic return follows a stable distribution, find the value at risk at the 95% level:

Compute the 95% value at risk point loss of the current S&P 500 index value, assuming the above distribution:

Find the expected shortfall of logarithmic return:

Compute the associated point loss:

A site has mean wind speed 7 m/s and Weibull distribution with shape parameter 2:

The resulting wind speed distribution over a whole year:

The power curve for a GE 1.5 MW wind turbine:

The total mean energy produced over the course of a year is then 4.3 MWh:

Estimate the distribution of the lengths of human chromosomes:

The expected chromosome length, given that the length is greater than the mean: