Probability Distributions and its basic definitions

Basic Definitions:

Random Variable: The Variable whose values are determined by chance is termed as random variable.

Probability Distribution: In this, the random variable values can suppose the corresponding probabilities of each.

Expected Value: It is the theoretical mean of variable.

Binomial Experiment: It is an experiment comprising of a fixed number of independent trials. Each trial can just contain two outcomes, or outcomes that can be decreased to two outcomes. The probability of each and every outcome should remain constant from trial to trial.

Binomial Distribution: It is the outcomes of a binomial experiment with their equivalent probabilities.

Multinomial Distribution: The probability distribution resultant from an experiment with a fixed number of independent trials is termed as multinomial distribution. Each trial consists of two or more mutually exclusive outcomes. The probability of each outcome should remain constant from trial to trial.

Poisson Distribution: The probability distribution employed if a density of items is distributed over a period of time. The sample size requires to be big and the probability of success to be very small.

Hypergeometric Distribution: When sampling is completed with no replacement, then the probability distribution of a variable with two outcomes is termed as Hypergeometric Distribution.

Probability Distributions:

Probability Functions: The probability function is a function that assigns the probabilities to values of a random variable.

•    All the probabilities should comprise between 0 to 1.
•    The sum of probabilities of the outcomes should be 1.

When such two conditions are not met, then the function is not a probability function. There is no necessity that the values of random variable only be between 0 to 1, only that the probabilities be between 0 to 1.

Probability Distributions:

We can make a probability distribution by listing all the random variables which can assume their corresponding probabilities.

It must be kept in mind that the random variable doesn’t mean that the values can be anything (that is, a random number). Random variables encompass a well defined set of outcomes and well defined probabilities for the occurrence of all outcomes. The random refers to the detail that the outcomes occur by chance -- that is, you do not know that outcome will occur subsequently.

Below is an example of probability distribution which results from the rolling of a single fair die.

x         1         2        3         4        5        6        sum
p(x)     1/6     1/6     1/6     1/6     1/6     1/6     6/6=1

Mean, Variance, and Standard Deviation:

Definitions for the population mean and variance employed with an ungrouped frequency distribution are:

μ = (Σxf/N) σ2 = [Σx2f – (Σxf)2/N]/N

Dividing by N is quite confusing. Remember that this is the population variance; the sample variance that was unbiased estimator for the population variance whenever it was divided by n-1.

By using the algebra, this is equivalent to:

μ = Σ x(f/N) σ2 = Σ x2(f/N) – (Σ x(f/N))2

Remember that the probability is a long term relative frequency. Therefore every f/N can be substituted by p(x). This can be stated as:

μ = xp(x) σ2 = Σ x2p(x) – [Σ xp(x)]2

It is much better that the last part of the variance is the mean squared. Therefore, the two formulas which we will be using are as:

μ = Σ xp(x) σ2 = Σ x2p(x) – μ2

Let’s see the example shown below:

x                 1         2        3        4        5        6        sum
p(x)            1/6     1/6     1/6     1/6     1/6     1/6     6/6 = 1
x p(x)         1/6     2/6     3/6     4/6     5/6      6/6     21/6 = 3.5
x^2 p(x)     1/6     4/6     9/6     16/6   25/6    36/6   91/6 = 15.1667

The mean is equal to 7/2 or 3.5

The variance is equal to 91/6 - (7/2)^2 = 35/12 = 2.916666..

Standard deviation is the square root of the variance = 1.7078

Doesn’t use round off values in an intermediate calculations. Just round off the final answer.

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