Definition with example:
The total set of all the probabilities of a random variable to attain all the possible values. Let me give an example. We toss a coin 3 times and try to find what the probability of obtaining head is? Here the event of getting head is known as the random variable. Now what are the possible values of the random variable, i.e. what is the possible number of times that head might occur? It is 0 (head never occurs), 1 (head occurs once out of 2 tosses), and 2 (head occurs both the times the coin is tossed). Hence the random variable is “getting head” and its values are 0, 1, 2. now probability distribution is the probabilities of all these values. The probability of getting 0 heads is 0.25, the probability of getting 1 head is 0.5, and probability of getting 2 heads is 0.25.
There is a very important point over here. In the above example, the random variable had 3 values namely 0, 1, and 2. These are discrete values. It might happen in 1 certain example that 1 random variable assumes 1 continuous range of values between x to y. In that case also we can find the probability distribution of the random variable. Soon we shall see that there are three types of probability distributions. Two of them deal with discrete values of the random variable and one of them deals with continuous values of the random variable.
Difference between probability and probability distribution: In probability we do not repeat the same event more than once. If the same event is repeated more than once, we calculate probability distribution. For example, to find the probability of getting head in tossing a coin once is an example of simple probability but when the coin is tossed more than once, we get probability distribution.
Here, the number of times the event occurs or is repeated is known as the number of trials and is represented by N.
...Important Discrete
ProbabilityDistributions
51
Chapter Goals
After completing this chapter, you should be able
to:
Interpret the mean and standard deviation for a
discrete probabilitydistribution
Explain covariance and its application in finance
Use the binomial probabilitydistribution to find
probabilities
Describe when to apply the binomial distribution
Use Poisson discrete probabilitydistributions to
find probabilities
52
Definitions
Random Variables
A random variable represents a possible
numerical value from an uncertain event.
Discrete random variables produce outcomes
that come from a counting process (e.g. number
of courses you are taking this semester).
Continuous random variables produce outcomes
that come from a measurement (e.g. your annual
salary, or your weight).
53
Definitions
Random Variables
Random
Variables
Ch. 5
Discrete
Random Variable
Continuous
Random Variable
Ch. 6
54
Discrete Random Variables
Can only assume a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 comes up
(then X could be 0, 1, or 2 times)
Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
55
ProbabilityDistribution for a
Discrete Random Variable
A probability...
...Learning Programmes Division
Second Semester 20102011
Course Handout
Course Number
Course Title
: AAOC ZC111
: Probability and Statistics
Course Email address : aaoczc111@dlpd.bitspilani.ac.in
Course Description
Probability spaces; conditional probability and independence; random variables and probabilitydistributions; marginal and conditional distributions; independent random variables, mathematical
exceptions, mean and variance, Binomial Poisson and normal distribution; sum of independent random
variables; law of large numbers; central limit theorem; sampling distributions; tests for mean using normal
and student’s distributions; tests of hypotheses; correlation and linear regression.
Scope and Objectives
At the end of the course, the student should be able to understand probabilistic & deterministic models and
statistical inference and apply these concepts to solve a variety of problems.
Prescribed Text Book
T1
Johnson Richard A. & C.B. Gupta, Miller & Freund’s Probability and Statistics for Engineers, PHI,
7th Ed., 2005.
Reference Books
R1.
Paul L. Meyer, Introductory Probability and Statistical Appl., Second Edition. AddisonWesley, 1970.
R2. M.S. Radhakrishnan, Probability & Statistics, DLPD Notes Note: Softcopy of this Supplementary notes will
be available for download from BITS DLP...
...Tutorial on Discrete ProbabilityDistributions
Tutorial on discrete probabilitydistributions with examples and detailed solutions.

Top of Form

Web  www.analyzemath.com 

Bottom of Form 
 Let X be a random variable that takes the numerical values X1, X2, ..., Xn with probablities p(X1), p(X2), ..., p(Xn) respectively. A discrete probabilitydistribution consists of the values of the random variable X and their corresponding probabilities P(X).
The probabilities P(X) are such that ∑ P(X) = 1Example 1:Let the random variable X represents the number of boys in a family.
a) Construct the probabilitydistribution for a family of two children.
b) Find the mean and standard deviation of X.Solution to Example 1: * a) We first construct a tree diagram to represent all possible distributions of boys and girls in the family. * Assuming that all the above possibilities are equally likely, the probabilities are:
P(X=2) = P(BB) = 1 / 4
P(X=1) = P(BG) + P(GB) = 1 / 4 + 1 / 4 = 1 / 2
P(X=0) = P(GG) = 1 / 4 * The discrete probabilitydistribution of X is given by X  P(X) 
0  1 / 4 
1  1 / 2 
2  1 / 4 
* * Note that ∑ P(X) = 1 * b) The mean µ of the random variable X...
...CHAPTER 3: PROBABILITYDISTRIBUTION
3.1
RANDOM VARIABLES AND PROBABILITYDISTRIBUTION
Random variables is a quantity resulting from an experiment that, by chance, can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable.
3.2
DISCRETE RANDOM VARIABLE
A random variable is called a discrete random variable if its set of posibble outcomes is countable. Probabilitydistribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. For example, the probabilitydistribution of rolling a die once is as below: Outcome, x Probability, P(x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6
The probabilitydistribution for P(x) for a discrete random variable must satisfy two properties: 1. The values for the probabilities must be from 0 to 1; 0 ≤ ( ) ≤ 1 2. The sum for P(x) must be equal to 1; ∑ ( ) = 1
QMT200
3.2.1 FINDING MEAN AND VARIANCE Mean of X is also referred to as its “expected value”.
= ( ) Where: = ∑[ ( )]
( )=
= (
) − [ ( )]
(
)=
[
( )] = ( )
Example 1 An experiment consists of tossing two coins simultaneously....
...The Poisson probabilitydistribution, named after the French mathematician SiméonDenis. Poisson is another important probabilitydistribution of a discrete random variable that has a large number of applications. Suppose a washing machine in a Laundromat breaks down an average of three times a month. We may want to find the probability of exactly two breakdowns during the next month. This is an example of a Poissonprobabilitydistribution problem. Each breakdown is called an occurrence in Poisson probabilitydistribution terminology. The Poisson probabilitydistribution is applied to experiments with random and independent occurrences. The occurrences are random in the sense that they do not follow any pattern, and, hence, they are unpredictable. Independence of occurrences means that one occurrence (or nonoccurrence) of an event does not influence the successive occurrences or nonoccurrences of that event. The occurrences are always considered with respect to an interval. In the example of the washing machine, the interval is one month. The interval may be a time interval, a space interval, or a volume interval. The actual number of occurrences within an interval is random and independent. If the average number of occurrences for a given interval is known, then by using the Poisson probability...
...ProbabilityDistribution Essay
Example Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable
Binomial Probability Function: it is a discrete distribution. Thedistribution is done when the results are not ranged along a wide range, but are actually binomial such as yes/no. This is used frequently in quality control, reliability, survey sampling, and other corporate and industrial situations. This type of distribution can measure levels of performance only if the results can be placed into a binomial order, such as with a point estimate where only one number is relied upon. For example, if you measure whether unit X had exceeded its monthly energy limits usage and is interested in a yes or no answer. This type of distribution gives the probability of an exact number of successes in independent trials (n), when the probability of success (p) on single trial is a constant.
The probability of getting exactly r success in n trials, with the probability of success on a single trial being p is:
P(r) (r successes in n trials) = nCr . pr . (1 p)(nr) = n! / [r!(nr)!]...
...APPLIED PROBABILITY AND STATISTICS
APPLIED PROBABILITY AND STATISTICS
DEPARTMENT OF COMPUTER SCIENCE
DEPARTMENT OF COMPUTER SCIENCE
STATISTICAL DISTRIBUTION
STATISTICAL DISTRIBUTION
SUBMITTED BY –
PREETISH MISHRA (11BCE0386)
NUPUR KHANNA (11BCE0254)
SUBMITTED BY –
PREETISH MISHRA (11BCE0386)
NUPUR KHANNA (11BCE0254)
SUBMITTED TO –
PROFESSOR
SUJATHA V.
SUBMITTED TO –
PROFESSOR
SUJATHA V.
ACKNOWLEDGEMENT –
ACKNOWLEDGEMENT –
First and foremost we like to thank our supervisor of the project Mrs Sujatha V. for her valuable guidance an advice. She inspired us greatly to work in this project. Her willingness to motivate us contributed majorly in this project. We also would like to thank her for showing us some example that related to our project.
Besides, we would also like to thank VIT University for providing us with a good environment and facilities to complete this project. Also, we would like to thank school of computer science (SCSE) of VIT University, for offering this subject and computing project. It has given us the opportunity to participate and learn about various methods of calculating statistical distribution.
Finally, an honourable mention goes to my team for completing this project. Without helps of particular mentioned above, we would have faced many difficulties while...
...random variable X be the number of people who buy a plain bagel. (a) Find the probabilitydistribution for X. (b) Suppose at least 3 people buy a plain bagel. What is the probability that exactly 4 people buy a plain bagel? 3. The probabilitydistribution for a discrete random variable X is given by the formula p(r) = for r = 1, 2, . . . , 6. (a) Verify that this is a valid probabilitydistribution. (b) Find P (X = 4). (c) Find P (X > 2). (d) Find the probability that X takes on the value 3 or 4. (e) Construct the corresponding probability histogram. 4. Two packages are independently shipped from Fort Collins, Colorado, to the same address in Seattle, Washington, and each is guaranteed to arrive within 4 days. The probability that a package arrives within 1 day is 0.10, within 2 days is 0.15, within 3 days is 0.25, and on the fourth day is 0.50. Let the random variable X be the total number of days it takes for both packages to arrive. Construct the probability histogram for X. 5. Suppose the random variable X has the probabilitydistribution given in the table below. r p(r) 2 0.15 3 0.25 5 0.15 7 0.10 11 0.30 13 0.05 r(r + 1) 112
(a) Find the mean, variance, and standard deviation of X. (b) Find the probability that X takes on a value smaller than the mean. 1
(c) Suppose the...