SPECIAL PROBABILITY DISTRIBUTION

Binomial Distribution

What is a binomial experiment?

A binomial experiment has the following characteristics:

The experiment involves repeated trials.
Each trial has only two possible outcomes - a success or a failure.
The probability that a particular outcome will occur on any given trial is constant.
All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution. It refers to the probabilities associated with the number of successes in a binomial experiment.

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution, as shown below.

Outcome, x	Binomial probability, P(X = x)	Cumulative probability, P(X < x)
0 Heads	0.125	0.125
1 Head	0.375	0.500
2 Heads	0.375	0.875
3 Heads	0.125	1.000

What is the number of trials?

The number of trials refers to the number of attempts in a binomial experiment. The number of trials is equal to the number of successes plus the number of failures.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. In this experiment, Heads would be classified as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY r successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X < 2); the probability of getting AT MOST 2 successes is indicated by P(X < 2); the probability of getting AT LEAST 2 successes is indicated by P(X > 2); the probability of getting MORE THAN 2 successes is indicated by P(X > 2).

Poisson Distribution

A Poisson distribution is the probability distribution that results from a Poisson experiment.

Attributes of a Poisson Experiment

A Poisson experiment is a statistical experiment that has the following properties:

The experiment results in outcomes that can be classified as successes or failures.
The average number of successes (μ) that occurs in a specified region is known.
The probability that a success will occur is proportional to the size of the region.
The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

Notation

The following notation is helpful, when we talk about the Poisson distribution.

e: A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.)
μ: The mean number of successes that occur in a specified region.
x: The actual number of successes that occur in a specified region.
P(x; μ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.

Poisson Distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. Theprobability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e^-μ) (μ^x) / x!

where x is the actual number of successes that result from the experiment, and eis approximately equal to 2.71828.

The Poisson distribution has the following properties:

The mean of the distribution is equal to μ .
The variance is also equal to μ .

Example

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

Solution: This is a Poisson experiment in which we know the following:

μ = 2; since 2 homes are sold per day, on average.
x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.
e = 2.71828; since e is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(x; μ) = (e^-μ) (μ^x) / x!
P(3; 2) = (2.71828^-2) (2³) / 3!
P(3; 2) = (0.13534) (8) / 6
P(3; 2) = 0.180

Thus, the probability of selling 3 homes tomorrow is 0.180 .

Cumulative Poisson Probability

A cumulative Poisson probability refers to the probability that the Poisson random variable is greater than some specified lower limit and less than some specified upper limit.

Example

Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

Solution: This is a Poisson experiment in which we know the following:

μ = 5; since 5 lions are seen per safari, on average.
x = 0, 1, 2, or 3; since we want to find the likelihood that tourists will see fewer than 4 lions; that is, we want the probability that they will see 0, 1, 2, or 3 lions.
e = 2.71828; since e is a constant equal to approximately 2.71828.

To solve this problem, we need to find the probability that tourists will see 0, 1, 2, or 3 lions. Thus, we need to calculate the sum of four probabilities: P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5). To compute this sum, we use the Poisson formula:

P(x < 3, 5) = P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5)
P(x < 3, 5) = [ (e^-5)(5⁰) / 0! ] + [ (e^-5)(5¹) / 1! ] + [ (e^-5)(5²) / 2! ] + [ (e^-5)(5³) / 3! ]
P(x < 3, 5) = [ (0.006738)(1) / 1 ] + [ (0.006738)(5) / 1 ] + [ (0.006738)(25) / 2 ] + [ (0.006738)(125) / 6 ]
P(x < 3, 5) = [ 0.0067 ] + [ 0.03369 ] + [ 0.084224 ] + [ 0.140375 ]
P(x < 3, 5) = 0.2650

Thus, the probability of seeing at no more than 3 lions is 0.2650.

Normal Distribution

Data can be "distributed" (spread out) in different ways. For example ;

But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: