P&SIA L1 C3 S4

Intro Stats: Islamic Approach -- Part 2: Probability and Statistics -

Lecture 1: Random Samples, Concept 3: Random Draws

S3: Random Variables

A Random Variable is a CHARACTERISTIC of an element of the population, chosen by a random draw. For example, we can choose a person at random from the population of Pakistan, and then ask about the HEIGHT, WEIGHT, AGE, INCOME, or any other characteristic. Let P be the population, and let W be the person chosen by a Simple Random Draw (SDR). Then any function X(W) is called a Random Variable -- X assigns to each W some characteristic of the person W.

Definition of Random Variable

Suppose W is a simple random draw from a population P and suppose function X(W) takes values 1,2,...,K.

Let E1 be the set of all elements W in P such that X(W)=1

Let E2 be the set of all elements W in P such that X(W)=2

...

Let EK be the set of all elements W in P such that X(W)=K

Then P(X(W)=J) = #EJ / #P, where #EJ is the number of elements in the set EJ and #P is the total number of elements in the population P.

Example of a Random Variable

Population of Pakistan = 132 million

W is a randomly chosen person, via a Simple Random Draw. This means that all people in Pakistan have equal chance 1/132million of being chosen.

DEFINE random variable X(W)=1,2,3,4 according to the Province of W, assigning 1 to Punjab, 2 to Sind, 3 to KPK and 4 to Balochistan. Then we have

P(X(W)=1 is 55.3%, P(X(W)=2 is 23.5%,

P(X(W)=3 is 15.9%, and P(X(W)=4) is 5,3%.

The above definition of a random variables is a COMPLETE description of it as a function from a population to characteristics of the population. In many applications, it is sufficient to use a SIMPLER description: A random variable X has OUTCOMES 1,2,3,... K with probabilities P1, P2, ...,PK. The probabilities must be percentages between 0 and1 and the sum of all the probabilities must be 100%.