Many studies are based on counts of the times a particular event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume, or any physical area in which there can be more than one occurrence of an event. Examples of variables that follow the Poisson distribution are the surface defects on a new refrigerator, the number of network failures in a day, the number of people arriving at a bank, and the number of fleas on the body of a dog. You can use the Poisson distribution to calculate probabilities in situations such as these if the following properties hold:
•You are interested in counting the number of times a particular event occurs in a given area of opportunity. The area of opportunity is defined by time, length, surface area, and so forth. •The probability that an event occurs in a given area of opportunity is the same for all the areas of opportunity. •The number of events that occur in one area of opportunity is independent of the number of events that occur in any other area of opportunity. •The probability that two or more events will occur in an area of opportunity approaches zero as the area of opportunity becomes smaller.
Consider the number of customers arriving during the lunch hour at a bank located in the central business district in a large city. You are interested in the number of customers who arrive each minute. Does this situation match the four properties of the Poisson distribution given earlier? First, the event of interest is a customer arriving, and the given area of opportunity is defined as a oneminute interval. Will zero customers arrive, one customer arrives, and two customers arrive, and so on? Second, it is reasonable to assume that the probability that a customer arrives during a particular oneminute interval is the same as the probability for all the other one minute intervals. Third, the arrival of one customer in any oneminute interval has...
...outcome xi according to its probability, pi. The mean also of a random variable provides the longrun average of the variable, or the expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X) is , formally defined by
Variance  The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation  The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
2. Define the following;
a) Binomial Distribution  is the discrete probabilitydistribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Therewith the probability of an event is defined by its binomial distribution. A success/failure experiment is also...
...The Poissonprobabilitydistribution, 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 Poissonprobabilitydistribution terminology. The Poissonprobabilitydistribution 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...
...The Poissondistribution is a discrete distribution. It is often used as a model for the number of events (such as the number of telephone calls at a business, number of customers in waiting lines, number of defects in a given surface area, airplane arrivals, or the number of accidents at an intersection) in a specific time period. It is also useful in ecological studies, e.g., to model the number of prairie dogs found in a square mile of prairie. The major difference between Poisson and Binomial distributions is that the Poisson does not have a fixed number of trials. Instead, it uses the fixed interval of time or space in which the number of successes is recorded.
Parameters: The mean is λ. The variance is λ.
[pic]
[pic] is the parameter which indicates the average number of events in the given time interval.
Ex.1. On an average Friday, a waitress gets no tip from 5 customers. Find the probability that she will get no tip from 7 customers this Friday.
The waitress averages 5 customers that leave no tip on Fridays: λ = 5.
Random Variable : The number of customers that leave her no tip this Friday.
We are interested in P(X = 7).
Ex. 2 During a typical football game, a coach can expect 3.2 injuries. Find the probability that the team will have at most 1 injury in this game.
A coach can expect 3.2 injuries : λ = 3.2....
...Probabilitydistribution
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 probabilitydistribution 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 probabilitydistribution of the random variable. Soon we shall see that there are three types of probabilitydistributions. Two of them deal with discrete values of the...
...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
UsePoisson 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...
...Probability and its Applications
Published in association with the Applied Probability Trust
Editors: S. Asmussen, J. Gani, P. Jagers, T.G. Kurtz
Probability and its Applications
Azencott et al.: Series of Irregular Observations. Forecasting and Model
Building. 1986
Bass: Diffusions and Elliptic Operators. 1997
Bass: Probabilistic Techniques in Analysis. 1995
Berglund/Gentz: NoiseInduced Phenomena in SlowFast Dynamical Systems:
A SamplePaths Approach. 2006
Biagini/Hu/Øksendal/Zhang: Stochastic Calculus for Fractional Brownian Motion
and Applications. 2008
Chen: Eigenvalues, Inequalities and Ergodic Theory. 2005
Costa/Fragoso/Marques: DiscreteTime Markov Jump Linear Systems. 2005
Daley/VereJones: An Introduction to the Theory of Point Processes I: Elementary
Theory and Methods. 2nd ed. 2003, corr. 2nd printing 2005
Daley/VereJones: An Introduction to the Theory of Point Processes II: General
Theory and Structure. 2nd ed. 2008
de la Peña/Gine: Decoupling: From Dependence to Independence, Randomly
Stopped Processes, UStatistics and Processes, Martingales and Beyond. 1999
de la Peña/Lai/Shao: SelfNormalized Processes. 2009
Del Moral: FeynmanKac Formulae. Genealogical and Interacting Particle
Systems with Applications. 2004
Durrett: Probability Models for DNA Sequence Evolution. 2002, 2nd ed. 2008
Ethier: The Doctrine of Chances. 2010
Feng: The Poisson–Dirichlet...
...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...
...Bernoulli and PoissonDistributions
The Binomial, Bernoulli and Poissondistributions are discrete probabilitydistributions in which the values that might be observed are restricted to being within a predefined list of possible values. This list has either a finite number of members, or at most is countable.
* Binomial distribution
In many cases, it is appropriate to summarize a group of independent observations by the number of observations in the group that represent one of two outcomes. For example, the proportion of individuals in a random sample who support one of two political candidates fits this description. In this case, the statistic is the count X of voters who support the candidate divided by the total number of individuals in the group n. This provides an estimate of the parameter p, the proportion of individuals who support the candidate in the entire population.
The binomial distribution describes the behavior of a count variable X if the following conditions apply: the number of observations n is fixed, each observation is independent and represents one of two outcomes ("success" or "failure") and if the probability of "success" p is the same for each outcome.
If these conditions are met, then X has a binomial distribution with parameters n and p, abbreviated B(n,p).
* Bernoulli...
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