Random Variable, Mathematical Expectation

Random Variables

When the population (or potential outcomes) are discrete, discrete random variables are created.

Example: After three coin tosses, we have

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

If the number of heads observed were the variable of interest, x, then possible events would include:

{x =0}= {TTT}

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

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

{x =3}= {HHH}.

The key concern is determining the likelihood of each of these events. Despite the fact that the sample space contains both heads and tails, keep in mind that X catch integer values.

We may find it more useful to numerically express a certain quality or attribute of experiment results in cases where our interest is not in the experiment's results per se. For instance, if there are three births, our intention might be to determine the likelihood that there will be three boys. Think about the 8 equally likely sample points in the sample space.

GGG GGB GBG BGG

GBB BGB BBG BBB

Look at the 'number of males out of three births' variable. This number can have values of 0, 1, 2, or 3, and it is random—given to chance. It fluctuates between sample locations in the sample space.

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

Any of the following may be a random variable:

• Discrete: If it only accepts a countable number of values For instance, the number of dots on a die, the proportion of heads in five coin tosses, the quantity of useful things, the proportion of girls in eight births, etc.
• Continuous: When it can have any value within a range of numbers (i.e. its possible values are infinite). As an illustration, measured information on heights, temperature, and time, etc.

A random variable has a probability law, which is a grammar that relates probabilities to the variable's various values.

The probability distribution of the random variable is sometimes referred to as the probability assignment. We use X to represent the random variable.

Mathematical Expectation

The expected value, sum, or integration of potential results from a random variable is known as a mathematical expectation. We also refer to it as the result of the likelihood of an event happening and the value associated with the event's actual observed occurrence.

The crucial characteristic of any random number is the anticipated outcome. E(x) stands for mathematical expectation, which is calculated using the formula below.

E(X)= Σ (x1p1, x2p2, …, xnpn)

Where,

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

P=probability of occurrences

n = number of all possible values.

When event A does not occur, an indicator variable's mathematical expectation can be zero (0), and when event A does occur, the indicator variable's mathematical expectation can be one (1). (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(∑aiXi)=∑ aiE(Xi). Here, ai, (i=1…n) are constants.

Example:

Consider the following table:

Determine the likelihood that the withdrawal ticket contains the number 1, assuming that each and every ticket will only have 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