Random Variables Lecture Lecturer : FATEN AL-HUSSAIN

Contents 4-1 Random Variables. 4-2 Discrete Random Variables. 4-3 Expected Value. 4-4 Expectation of a Function of a Random Variables. 4-5 Variance. 4-6 The Bernoulli and Binomial Random Variables. Summary Problems Theoretical Exercises Self –Test Problems and Exercises.

4.3 Expected Value If X is a discrete random variable with probability distribution function P(X=x) then the expectation of X, written as E(X) is defined as

Let X denote a random variable that takes on any of the values −1, 0, and 1 with respective probabilities P{X = −1} =.2 P{X = 0} =.5 P{X = 1} =.3 Compute E[X 2 ]. x01 P(x).2.5.3

Expectation of general / derived function

Definition If X is a random variable with mean μ, then the variance of X, denoted by Var(X), is defined by Var(X) = E [(X − μ) 2 ]

Calculate Var(X) if X represents the outcome when a fair die is rolled

Variances for general / derived function

Look at the experiment which has only two outcomes. such as flipping a coin. Where the possible outcome is either head or tail. Any experiments which has only two outcomes is termed as Bernoulli trials. Examples of Bernoulli trials:  Select one student at random and determine their sexes  Throw a dice and determine the outcome, odd or even  In these experiments, one outcome is termed as success and the other is a failure.  The success is when the event is occurs and failure when it’s not. The probability of success is denoted by p and failure by 1-p = q.

Bernoulli Random P(X)= P for x=1 (success) 1-P for x=0 (failure)

If the Bernoulli trials is repeated n times and the number of success is recorded. The random variable with these number of success is having a Binomial distribution. Example of Binomial distribution:  A fair coin is tossed 10 times and number of head observer is recorded. The random variable in this example is number of head observed.  A fair dice is thrown 5 times and the number of times face showing 6 is observed. The random variable in this case is the number of times 6 is observed.

Binomial Random P(X)= P for x=1 (success) 1-P for x=0 (failure)

A coin is tossed 3 times. find the probability mass function of the number of heads obtained.

A fair dice is tossed 4 times. find the mean, variance and slandered deviation of obtaining the number 6.

If X is discrete random variable which represents the number of times random events occurs in an interval of time on in an interval of space, than X is a Poisson random variable. The number of events occurs is termed as success. Examples of Poisson random variables are.  the number of accidents occurring on certain highway in one month.  the number of telephone calls received from 9.00am to am  the number of misspelled words in one page  the number of bacteria in one liter of water