Expected

Question 1

Expectation for discrete variables

Answer 1

The expected value of a random variable X, E(X), is a measure of its central tendency.
For a discrete random variable X with PMF p(x), E(X) is defined as a sum, over all possible values x, of the quantity x*p(x).
E(X) represents the center of mass of a collection of locations and weights, {x, p(x)}.

Another term for expected value is mean

The mean characterizes the central tendency of the
| distribution.

Question 2

One of the nice properties of the expected value operation is that it’s linear

Answer 2

This means that, if c is a constant, then E(cX) = c*E(X). Also, if X and Y are two
| random variables then E(X+Y)=E(X)+E(Y). It follows that E(aX+bY)=aE(X)+bE(Y).

Question 3

Expectation for continuous random variables

Answer 3

for continuous random variables, E(X) is the area under the function t*f(t), where f(t) is the PDF (probability density function) of X.

Question 4

apply(allsam, 1, mean)

Answer 4

We simply call apply with
| the arguments allsam, 1, and mean. The second argument, 1, tells ‘apply’ to apply the third argument ‘mean’ to the rows of the matrix.

Question 5

sample mean and population mean

Answer 5

The expected value or mean of the sample mean is the population mean. What this means is that the sample mean is an unbiased estimator of the
| population mean.

The more data that goes into the sample mean, the more concentrated its density / mass function is around the population mean.