Download Examples of Poisson random variables are.

Random Variables
Lecturer : FATEN AL-HUSSAIN
Lecture
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].
x
P(x)
-1
.2
0
.5
1
.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)=
1-P
for x=0 (failure)
P
for x=1 (success)
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)=
1-P
for x=0 (failure)
P
for x=1 (success)
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 10.00 am
the number of misspelled words in one page
the number of bacteria in one liter of water