The Binomial, Bernoulli and Poisson distributions are discrete probability distributions in which the values that might be observed are restricted to being within a pre-defined 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 distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jacob Bernoulli, is a discrete probability distribution The distribution of a discrete random variable taking two values, usually 0 and 1. An experiment or trial that has exactly two possible results, often classified as 'success' or 'failure', is called a Bernoulli trial. If the probability of a success is p and the number of successes in a single experiment is the random variable X, then X is a Bernoulli variable (also called a binary variable) and is said to have a Bernoulli distribution with parameter p. A binomial variable with...

...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 Binomialdistributions 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.
Random Variable :...

...useful distribution for ﬁtting discrete data: revival
of the Conway–Maxwell–Poissondistribution
Galit Shmueli,
University of Maryland, College Park, USA
Thomas P. Minka and Joseph B. Kadane,
Carnegie Mellon University, Pittsburgh, USA
Sharad Borle
Rice University, Houston, USA
and Peter Boatwright
Carnegie Mellon University, Pittsburgh, USA
[Received June 2003. Revised December 2003]
Summary. A useful discretedistribution (the Conway–Maxwell–Poissondistribution) is revived
and its statistical and probabilistic properties are introduced and explored. This distribution is a
two-parameter extension of the Poissondistribution that generalizes some well-known discrete
distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions
derived from these discrete distributions (i.e. the binomial and negative binomialdistributions).
We describe three methods for estimating the parameters of the Conway–Maxwell–Poissondistribution. The ﬁrst is a fast simple weighted least squares method, which leads to estimates
that are sufﬁciently accurate for practical purposes. The second method, using maximum likelihood, can be used to reﬁne the initial estimates. This...

...The BinomialDistribution
October 20, 2010
The BinomialDistributionBernoulli Trials
Deﬁnition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure.
1
Tossing a coin and considering heads as success and tails as failure.
The BinomialDistributionBernoulli Trials
Deﬁnition ABernoulli trial is a random experiment in which there are only two possible outcomes - success and failure.
1
Tossing a coin and considering heads as success and tails as failure. Checking items from a production line: success = not defective, failure = defective.
2
The BinomialDistributionBernoulli Trials
Deﬁnition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure.
1
Tossing a coin and considering heads as success and tails as failure. Checking items from a production line: success = not defective, failure = defective. Phoning a call centre: success = operator free; failure = no operator free.
2
3
The BinomialDistributionBernoulli Random Variables
A Bernoulli random variable X takes the values 0 and 1 and P(X = 1) = p P(X = 0) = 1 − p. It can be easily checked that the mean...

...The Poisson probability distribution, named after the French mathematician Siméon-Denis. Poisson is another important probability distribution 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 Poisson probability distribution problem. Each breakdown is called an occurrence in Poisson probability distribution terminology. The Poisson probability distribution 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...

...POISSONDISTRIBUTION
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 Poissondistribution 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 Poissondistribution 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...

...#1 True or false: Even if the sample size is more than 1000, we cannot always use the normal approximation to binomial.
Solution:
If a sample is n>30, we can say that sample size is sufficiently large to assume normal approximation to binomial curve.
Hence the statement is false.
#2
A salesperson goes door-to-door in a residential area to demonstrate the use of a new Household appliance to potential customers. She has found from her years of experience that after demonstration, the probability of purchase (long run average) is 0.30. To perform satisfactory on the job, the salesperson needs at least four orders this week. If she performs 15 demonstrations this week, what is the probability of her being satisfactory? What is the probability of between 4 and 8 (inclusive) orders?
Solution
p=0.30
q=0.70
n=15
k=4
[pic]
Using Megastat we get
| | |15 |
| |0.3 | P |
| | | |
| | |Cumulative |
|k |p(k) |Probability |
|0 |0.00056 |0.00056 |
|1 |0.00503 |0.00559 |
|2 |0.02154 |0.02713 |
|3 |0.05848 |0.08561 |
|4 |0.11278 |0.19838 |
|5 |0.16433 |0.36271 |
|6 |0.18781 |0.55052...

...Probability distribution
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...

...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) BinomialDistribution - is the discrete probability distribution 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 binomialdistribution. A success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomialdistribution is a Bernoullidistribution. The binomialdistribution is the basis for the popular binomial test of statistical significance. The binomialdistribution is frequently used to model the...

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