STA642 Midterm Slides Lectures 1-107 by Mahar Afaq

Virtual University of Pakistan · STA642 · Distributions · by · Dr. Saleha Naghmi Habibullah · Topic No. 1 · Random · Experiment, · Outcomes, Sample Space · & Events · Random Experiment

Any experiment or process in which the outcome is unpredictable that is eligible to be called a random experiment.

Example 1:

Tossing of a Coin => a Head or

a Tail.

However, until it lands on the floor, we do not know which of these two will turn up.

Example 2:

Survey of the Employees of an Organization:

Question on Marital Status:

Possible outcomes: single, married, separated, divorced, widowed

However, until we obtain a reply from the respondent, we do not know what is his/her marital status.

Random Experiment

Any experiment or process in

which the outcome is unpredictable that is eligible to be called a random experiment.

Outcomes

The various possible results of a random experiment.

Example 1: Head, Tail

Example 2: Single, Married, Separated, Divorced, Widowed

Sample Space

It is the set of all possible outcomes of a random experiment.

Example 1: Head , Tail Example 2: Toss a single die 1,2,3,4,5,6} Sample space will be the set of all of these taken together written as S= {1,2,3,4,5,6}

In case of two fair dice, the sample space consists of the 36 ordered pairs:

(6,6)}.

Event

Any subset of a sample space is called an Event.

Example 1

The set of ordered pairs which yield a sum which is an even number

denoted as A= {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), ,(6,6)}.

Example 2

The set of ordered pairs in which

both the numbers are same

denoted as B= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

A= {(1,1), (1,3), (1,5), (2,2),

B= {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

Event B is a subset of event A.

Topic No. 02 · Definition of · Concept of a Random Variable

It is that variable which is associated with a random experiment.

Example:

Let the random experiment be the toss of a fair coin and the possible outcomes are Head and Tail. And the sample space associated with the experiment denoted by C be C = {H,T}, where H and T represent heads and tails, respectively.

Let X be a function such that X(T) = 0 and X(H)=1. Thus X is a realvalued function defined on sample space C.

That X is a function defined on the sample space C, X(T) = 0 and X(H)=1 which takes us from the sample space C to a space of real numbers D= {0,1}.

C is H and T and applying the function the image we obtain i.e the domain and range concept, So, if the domain here is H and T then the range will be 0 and 1.

Consider a random experiment with a sample space C. A function X, which assigns to each element c C one and only one number X(c) = x, is called a random variable.

The space or range of X is the set of real numbers D = { x : x = X(c), c C}.

Given a random variable X, its range D becomes the space of interest. Besides inducing the space D, X also induces a probability which we call the distribution of X.

We are interested in knowing the probability of getting a 0 and 1, and as we already know that if a fair coin then the probability of getting a 0 is half and also for the probability of getting a 1 is half as well. But, generally in this case, we can say that the set D is a finite or countable set or an interval of real numbers.

When a set is finite then, we are referring to discrete random variables. We call random variables of the first type discrete random variables, while those of the second type continuous random variables.

Discrete random variable has two situation, one is exactly the same I mentioned earlier where a set D is a finite set that has the elements {0, 1} or the other, if we toss a fair die we have the elements{1,2,3,4,5,6}.

The second situation ,which also gives you a discrete random variable that is when it is the Countably infinite set.

Example:

D is the set consisting of the numbers 1,2,3, and so on, which mean that they are countable but not finite. So, in this situation, we say that this is a Discrete random variable or if they are finite the elements may be called d1,d2 m then also it is a discrete random variable.

Topic No. 3

Example:

Let a card be selected from an ordinary deck of playing cards. The outcome is one of these 52 cards. Suppose that P assigns a probability of 1/52 to each outcome c.

Let X(c)=4 if c is an ace, let X(c)=3 if c is a king, let X(c)=2 if c is a queen, let X(c)=1 if c is a jack and let X(c)=0 otherwise. Now, we can develop a table with two columns.

The first column consisting of the values of the random variable X i.e. 0,1,2 , 3,4 and the second are the probabilities.

The classical definition of probability i.e. favorable over the total. The sum of probability should be equal to 1.

Topic No. 04 · Variable · (Discrete Random Variable)

We say a random variable is a discrete random variable if its space is either finite or countable.

Example:

Consider a sequence of independent flips of a coin, each resulting in a head (H) or a tail (T). Moreover, on each flip, we assume that H and T are equally likely; that is, P(H)=P(T)=1/2.

This is that case, when the space is Finite.

x · p(x) · sum