Obtain the raw moments of a continuous distribution:

Obtain the mean of a discrete distribution:

Obtain the variance of a truncated distribution:

Construct a mixture density, here a Poisson-inverse Gaussian mixture:

Obtain the same result directly using

ParameterMixtureDistribution:

Verify Jensen inequality

for a concave function

and a lognormal distribution:

An insurance policy reimburses a loss up to a benefit limit of 10. The policy holder'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 three such claims will be made. Find 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:

Under an excess of loss reinsurance agreement, a claim is shared between the insurer and reinsurer only if the claim exceeds a fixed amount, called the retention level. Otherwise, the insurer pays the claim in full. Compute the expected value of the amounts,

and

, paid by the insurer and the reinsurer for a retention level of

if the claims follow a lognormal distribution with parameters

and

. Find the expected insurer claim payouts:

Find the expected reinsurer payouts to the insurer:

Compute the expected time value of a death benefit of $1 paid at time

, where

is drawn from a Gompertz-Makeham distribution:

Find the annual premium, which is usually paid at the beginning of a policy year, that is necessary to make the expected time value of that payment stream for

periods (where

is drawn from a Gompertz-Makeham distribution) equal to the net single premium:

The resulting net annual premium:

The fractional change of stock price

at time

(in years) is assumed lognormally distributed with parameters

and

:

Compute the expected stock price at epoch

:

Assuming an investor can invest money for a year at a continuously compounded yearly rate

risk-free, the risk-neutral pricing condition requires:

Solve for parameter

:

Consider an option to buy this stock a year from now, at a fixed price

. The value of such an option is:

The risk-neutral price of the option is determined as the present value of the expected option value:

Assuming rate

of 5%, volatility parameter

of 0.087, an initial price of $200 per share of stock, and a strike price of $190 per share, the Black-Scholes option price is:

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

Find the mean time to failure (MTTF) for an exponential life distribution:

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:

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 player bets amount

in a casino with no betting limit in a game with a chance of winning

. If he loses he doubles the bet, and if he wins he quits, hence the number of games played follows a geometric distribution, with expected number of games played represented as follows:

The cash reserve needed to win the

game:

The player always leaves the casino collecting the amount of the initial bet:

The cash reserve needed to execute the above strategy is finite only for strictly favorable games, where

:

A drug has proven to be effective in 40% of cases. Find the expected number of successes when applied to 700 cases:

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

Find the mean if the signal-to-noise ratio has a Weibull distribution: