Random variable and mathematical expectation

Introduction Random variable and mathematical expectation

• A variable whose numerical value is determined by the outcomes of a random experiment is called random variable.
• It is also known as a chance or stochastic variable.
• The values of random variables are denoted by capital letters and their particular values are generally denoted by small letters.
• For example; consider a random experiment of tossing two coins. Then the possible outcomes are {HH, HT, TH, TT}.
• Mathematical expectation is simply an average value of random variables. More specially, it is the weighted average of the random variable.

Let X be a variable whose value is associated with the number of getting head in a coin toss experiment then,

Where, 0 indicates, no head; 1 indicates, occurrence of one head; 2 indicates, occurrence of two heads. Thus, the random variable takes the values 0,1 or 2.Therefore, we can say that X is a random variable.

Types of random variable

• Continuous random variable.
• Discrete random variable.

What is Discrete random variable?

• If the random variable takes only the discrete values ie exact or countable or integers or non-fractional values then it is called discrete random variable.
• No. of getting head in a coin tossed experiment, no. of goals in a football match, no. of students in a class etc are some example of discrete random variables.

What Continuous random variable?

• A random variable that takes any value fractional, non-fractional, uncountable, integers or non-integers within a specified range is called continuous random variable. For example; amount of rainfall, temperature, height and weight of the people etc.
• More precisely, the random variables which can be measured by counting process are called discrete random variables and the random variables which can be measured by using instruments are called continuous variables.

Characteristics of Random Variable

• If X and Y be two random variables and a & b be any two constants then,
• aX, bY, aX+b, aX+bY are also random variables.
• X+Y, X-Y, XY, X/Y are also random variables.
• 1/X, X2 are also random variables etc.

Mathematical Expectation of discrete, continuous random variable

Mathematical expectation is simply an average value of random variable. More specially, it is the weighted average of the random variable.

Mathematical Expectation of Discrete Random Variable

• The expected value of discrete random variable is a weighted average of all the possible values of random variables where the weights are the probabilities associated with the corresponding values.
• Symbolically, let X be a discrete r. v. which can take n values 𝑥𝑖= ( i = 1, 2, … , n) with respective probabilities pi, then the expectation of X is given by,

$E\left(X\right)=\frac{P_1{x}_1+P_2{x}_2+P_3{x}_3+····+P_n{x}_n}{P_1+P_2+P_3+···+P_n}$

$=\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}} =\overline{\mathrm{X} } \because \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{i}}=P_1+P_2+P_3+···+P_n=1$

$or\mathit{,}E\left(X\right)=\sum x.P\left(x\right)$

• Thus, the mathematical expectation can also be defined as the sum of the product of the different random variables and their corresponding probabilities

Mathematical Expectation of Continuous Random Variable

If X is a continuous r. v. with probability density function f(x), then the expected value of X is given by,

$E\left(X\right)={\int }_{–\infty }^{+\infty }x.f\left(x\right)dx ; –\infty \le x\le +\infty$

Properties of Expectation

Addition theorem of Expectation:
The expectation of the sum of the number of random variables is equal to the sum of their individual expectations.
Symbolically, if X and Y are two random variables then
E (X+Y) = E (X) + E(Y).

Multiplication theorem of Expectation:
The expectation of product of a number of independent random variables is equal to the product their individual expectations.
Symbolically, if X and y are two independent random variables then,
E (XY) = E (X) . E (Y). If X is a random variable and C is a constant, then,
i. E (C) = C
ii. E (CX) = C E (X)
iii. E (X+C) = E (X) + C
iv. If X ≥ 0 then E (X) ≥ 0 etc.

Moments

Raw moments: The rth moment about origin (raw moment with A = 0) of the random variable X is given by

Where, r = 1, 2, 3, 4.

Central moments: The rth moment about mean of the random variable X is given by:

• Where, r = 1, 2, 3, 4.
• Remark:
• Central moments can also be calculated by using the
• relationship between raw moments and central moments.

Variance: Variance of the random variables is generally expressed in terms of expectation. Let X be a random variable then the variance of X, denoted by Var.(X) or V(X) is defined as:

Properties of Variance: Let X and Y be two variables and a & c be any constants, then

• V (c ) = 0
• V (cX) = c2 V (X).
• V ( a + X ) = V (X).
• V (a + bX) = b2 V (X).
• V (aX ± bY) = a2 V (X) + b2 V (Y) ± ab .Cov.(X, Y). etc.

Covariance of two random variables

• The covariance of two random variables X and Y is given by
• Cov(X,Y) = E[{X-E(X)}.{Y-E(Y)}]
• C0v(X,Y) = E(XY) – E(X).E(Y)
• Remark:
• If X and Y are two independent random variables then
• C0v(X,Y) = E(XY) – E(X).E(Y)
• = E(X).E(Y) – E(X).E(Y)
•  ∴ C0v(X,Y) = 0
• Cov(X,X) = V(X)
• Cov(Y,Y) = V(Y)

Conditional mathematical Expectation

Let X and Y be any two random variables then the conditional expectation of X given Y is given by

Conditional variance of random variable:

Let X and Y be two random variables then the conditional variance of X given Y is given by:

Similarly, the conditional variance of Y given X is given by

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