random variable;mathematical expectation

Random Variables

The discrete random variables arises in situations when the population (or possible outcomes) are discrete.

Example: Toss a coin 3 times, then we have

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

Let the variable of interest, x, be the number of heads observed then possible events would be

{x =0}= {TTT}

{x =1}= {HTT, THT, TTH}

{x =2}= {HHT, HTH, THH}

{x =3}= {HHH}.

The important question is to ï¬Ând the probability of each these events. Remember that X catch integer values even though the sample space consists of heads and tails.

In many situations, our interest does not lie in the outcomes of an experiment as such; we may find it more useful to describe a particular property or attribute of the outcomes of an experiment in numerical terms. For example, out of three births; our intension may be in the matter of the probabilities of the number of boys. Consider the sample space of 8 equally likely sample points.

GGG GGB GBG BGG

GBB BGB BBG BBB

Let’s see at the variable ‘number of boys out of three births’. This number varies among sample points in the sample space and can take values 0,1,2,3, and it is random –given to chance.

A random variable is an uncertain quantity whose outcome depends on chance.

A random variable may be in the following:

• Discrete: when it takes only a countable number of values. For instance, number of dots on dice, number of heads in five coin tossing, number of good items, number of girls in eight births, etc.
• Continuous: when it can take on any value in an interval of numbers (i.e. its possible values are infinite). For example, measured data on heights, temperature, and time, etc.,

A random variable has a probability law - a syntax that assigns probabilities to different values of the random variable.

The probability assignment is known as the probability distribution of the random variable. We denote the random variable by X.

Mathematical expectation

Mathematical expectation is the expected value, summation or integration of possible outcomes from a random variables. We also call it as product of probability of an event occurring and the value related with the actual observed occurrence of the event.

The expected result is the important property of any random number. Mathematical expectation is denoted by E(x) and is calculated by the formula given below,

E(X)= Σ (x₁p₁, x₂p₂, …, x_np_n)

Where,

X=random variable with the probability function f(x)

P=probability of occurrences

n = number of all possible values.

If there is no occurrence of event A then the mathematical expectation of an indicator variable can be zero (0)and if there is an occurrence of an event A then the mathematical expectation of an indicator can be one (1).

Properties of mathematical expectation:

• The expected value of the sum of the two variables is equal to the sum of the expected mathematical value of X and the mathematical expectation of Y given that X and Y are two random variables, provided that the mathematical expectation exist, i.e. E(X + Y)= E(X) + E(Y).
• The expected value of the product of the two random variables is equal to the product of the expected mathematical value of that two variables, given that the two variables are independent in their nature, i.e. E(XY)=E(X)E(Y).
• The expected value of the product of a constant and function of random variable is equal to the product of the product of the constant and the mathematical expectation of the function of that random variablee. E(a *f(X))=a E(f(X))where “a” is a constant and f(x) is a function.
• The expected value of the sum of product of a constant and a function of a random variable and the other constant is always equal to sum of product of the constant and the expected mathematical value of the function of that random variable and other constant, provided that theirs mathematical expectation exist. i.e. E(aX+b)=aE(X)+b where, a and b are constant.
• The expected mathematical value of the linear combination of random variables is always equal to the sum of the product between mathematical expectation of n number of variables and n constant, i.e. E(∑a_iX_i)=∑ a_iE(X_i). Here, a_i, (i=1…n) are constants.

Example:

Consider the following table:

Calculate the probability that the withdrawal ticket is of 1, given that each and every ticket will contain one and only one number.

Solution:

Let tickets = x

No of tickets = f

Now,

Therefore, the mathematical expectation of getting 1 is,

E(x₁)= ∑x₁.P(x₁)

=1*0.1

=0.1

(References)

Chaudary, A.K. (2061).Business statistics. kathmandu:Bhundipuran Prakshan

Dhakal Bashanta (2014).Business Statistics,Buddha academic publisher

Sthapit, Azaya Bikram(2006),Business Statistics,Asmita publication