The **expected value** (or mean) the X, whereby X is a discrete random variable, is a weighted median of the feasible values that X have the right to take, each value being weighted follow to the probability the that event occurring. The supposed value of X is generally written together E(X) or m.

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E(X) = S x P(X = x)

So the supposed value is the sum of: <(each that the possible outcomes) × (the probability of the outcome occurring)>.

In much more concrete terms, the expectation is what you would suppose the outcome of one experiment to be on average.

**Example**

What is the intended value when we role a fair die?

There room six feasible outcomes: 1, 2, 3, 4, 5, 6. Each of these has actually a probability that 1/6 of occurring. Allow X stand for the result of the experiment.

Therefore P(X = 1) = 1/6 (this means that the probability the the outcome of the experiment is 1 is 1/6)P(X = 2) = 1/6 (the probability that you litter a 2 is 1/6)P(X = 3) = 1/6 (the probability the you litter a 3 is 1/6)P(X = 4) = 1/6 (the probability the you throw a 4 is 1/6)P(X = 5) = 1/6 (the probability that you litter a 5 is 1/6)P(X = 6) = 1/6 (the probability that you throw a 6 is 1/6)

E(X) = 1×P(X = 1) + 2×P(X = 2) + 3×P(X = 3) + 4×P(X=4) + 5×P(X=5) + 6×P(X=6)

Therefore E(X) = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 = 7/2

So the expectation is 3.5 . If friend think about it, 3.5 is halfway in between the feasible values the die have the right to take and also so this is what girlfriend should have actually expected.

**Expected worth of a duty of X**

To discover E< f(X) >, where f(X) is a role of X, usage the complying with formula:

E< f(X) > = S f(x)P(X = x)

**Example**

For the over experiment (with the die), calculation E(X2)

Using ours notation above, f(x) = x2

f(1) = 1, f(2) = 4, f(3) = 9, f(4) = 16, f(5) = 25, f(6) = 36P(X = 1) = 1/6, P(X = 2) = 1/6, etc

So E(X2) = 1/6 + 4/6 + 9/6 + 16/6 + 25/6 + 36/6 = 91/6 = 15.167

The expected value that a consistent is just the constant, so for example E(1) = 1. Multiply a random variable through a constant multiplies the intended value by that constant, therefore E<2X> = 2E

A beneficial formula, whereby a and b space constants, is:

E

**Variance**

The variance that a random variable tells us something around the spread of the possible values of the variable. For a discrete arbitrarily variable X, the variance of X is created as Var(X).

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Var(X) = E< (X – m)2 > whereby m is the intended value E(X)

This can additionally be written as:

Var(X) = E(X2) – m2

The * typical deviation* of X is the square root of Var(X).

Note the the variance does no behave in the same way as expectation when we multiply and add constants to arbitrarily variables. In fact:

Var

You is because: Var

= E< a2X2 + 2abX + b2> - (aE(X) + b)2= a2E(X2) + 2abE(X) + b2 - a2E2(X) - 2abE(X) - b2= a2E(X2) - a2E2(X) = a2Var(X)