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Expected Value and Variance


Expected Value

Central Tendency
The value at which the values or pdf of a random variable tend to accumulate or cluster. It is usually described as the average value.

One of the ways to describe the pdf of a random variable is with a measure of central tendency. One of these measures is called the expected value. For a discrete random variable the expected value can be defined as:

E(X) = μX = ∑all kk * pX(k)

Where k is a value of a random variable X.

For a continuous random variable the expected value is defined as:

E(y) = μY = ∫
∞
-∞
y * f(y)dy

Where y is any value of random variable Y.

Example:

if X has the following pdf in graph form:

statistics and probability binomial distribution

the mean or expected value can be found as follows:

E(X) = μX =  0 *  1
8 		 + 1 *  3
8 		 + 2 *  3
8 		 + 3 *  1
8
         = 12
8
         = 1 1
2

Properties:

For linear functions:

E( aX + bY ) = aE(X) + bE(Y)

For non linear functions:

For discrete random variables:

E(g(X)) = ∑all kg(k)*pX(k)

For continuous random variables:

E(g(Y)) = ∫
∞
-∞
g(y)*fY(y)dy

For independent random variables:

E(XY) = E(X)*E(Y)

Variance

When the amount of dispersion of the random variable needs to be known, a value called variance is used. The variance measures how far apart the values of the random variable are. This is done by taking the difference of the values of the random variables and the mean. Each difference is squared to make it positive. All of the squared differences are then added together and divided by the number of differences. The variance is usually calculated with either the number of differences or the number of differences minus one. The square of the differences is used because adding up the differences could cause the sum to equal 0 and this would not be an accurate description of the amount of dispersion. One of the formulas for calculating variance is as follows:

Var(x) = σ2 =   1
n 		∑ni = 1( xi - μX )2

The variance formula has other forms also:

The discrete form is:

var(X) = σ2 = E(( X - μX )2 )
= ∑all k( k - μX )2pX(k)

For continuous random variables, it is:

var(Y) = σ2 = E((Y - μ)2 ) = ∫
∞
-∞
( y - μ )2fY(y)dy

also,

var(W) = E(W2) - μ2

Some of the properties of variance are as follows:

For linear functions:

var( aX + b ) = a2var(X)

also:

var( X + Y ) = var(X) + var(Y)

Standard Deviation

Sometimes the spread factor is described in terms of standard deviation. This is just the square root of variance.

standard deviation = σX = √&sigmax2 = √var(X)

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