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Random Variable:

A random variable is a numerical description of the outcome of a statistical experiment. ... For a discrete random variable, x, the probability distribution is defined by a probability mass function, denoted by f(x). This function provides the probability for each value of the random variable.

For example

If a coin tossed three times then sample space will be

S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

Getting head will be a Random variable(X), and its possible value will be x = 0,1,2,3.

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Define Expectation:

It is also known as the expected value,denoted by E(x),is the summation(for discrete variable) or integration(for continuous varaible) of a possible values from a random variable. It is also known as the product of the probability of an event occurring, denoted P(x), and the value corresponding with the actual observed occurrence of the event.

The mathematical expectation will be given by the mathematical formula as,

E(X)= (x₁ p₁ + x₂ p₂ + x₃ p₃ …+ x_n p_n)

or

E(X)= ∑x_i p_i

where x is a random variable with the probability function, f(x), p is the probability of the occurrence, and n is the number of all possible values.

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Properties of Expectation

(i) . If X and Y are random variables, then E(X + Y) = E(X) + E(Y).

(ii) . If X₁, X₂, … , X_n are random variables , then E(X₁ + X₂ + … + X_n) = E(X₁) + E(X₂) + … + E(X_n) = ∑_i E(X_i).

(iii) . For random variables, X and Y, E(XY) = E(X) E(Y). Here, X and Y must be independent.

(iv) . If a is any constant and X is a random variable, E[aX] = a E[X] and E[X + a] = E[X] + a.

(v) . For any random variable, X > 0, E(X) > 0.   (because X is positive and probability cannot be nagative)

(vi) . E(Y) ≥ E(X) if the random variables X and Y are such that Y ≥ X.

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(vii) . The expected value of the constant value is constant itself .E(k) = k

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Variance of Random Variables

The variance of a random variable shows the variability or the dispersion of the random variables. It shows the distance of a random variable from its mean. It is calculated as SD_x² = Var (X) = ∑ (x_i - µ)² p(x_i) = E(X - µ)² or, Var(X) = E(X²) - [E(X)]².

E(X²) = ∑ x_i² p(xi), and [E(X)]² = [∑x_i p(xi)]² = µ².

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Properties of Variance of Random Variables

(i)   The variance of any constant is zero i.e, V(k) = 0, where k is any constant.

(ii)   If X is a random variable, and a and b are any constants, then V(aX + b) = a² V(X).

(iii)   For any pair-wise independent random variables, X₁, X₂, … , X_n and for any constants a₁, a₂, … , a_n;

V(a₁X₁ +a₂X₂ + … +a_nX_n ) = a₁² V(X₁) +a₂² V(X₂) + … + a_n² V(X_n).