How can I find the expected value of a random variable using the Statistics Toolbox 7.0 (R2008b) given its distribution and a constraint?

I would like to find the expected value of a random variable given a distribution and a constraint. For example, given a normal distribution, what is the expected value of x with the constraint that x > 0.

The unconditional expected value would be x = 0.

Answer 1

[MathWorks Support Team]

MATLAB is an excellent tool for this type of calculation .

To calculate the expected value of x given x>0 and a normal distribution, you can use two methods. First, you can simulate the data set and then select the mean of only the values of x which satisfy the constraint. For example:

Generate some normal data, and compute the unconditional mean

x = randn(10000,1);

mean(x)

 ans =

   5.5206e-004

Compute the conditional mean by simulation. Generate data from the conditional distribution and take the mean.

mean(x(x>0))

 ans =

     0.7955

The second method is to use a numerical computation of the expected value over the conditional distribution. This conditional distribution has the normal pdf over the region above 0, scaled by 1 minus the cdf evaluated at 0. The integral should go to +Inf, but I know the probability is very small for high values so I stop at 10.

quadgk(@(x) x.*normpdf(x)./(1-normcdf(0)), 0, 10)

 ans =

     0.7979

The second method is extendable to many of the distributions that MATLAB supports.