Probability

The theory of probability deals with measurement of the chances of a event to occur out of a set of all possible outcomes.  The probability is defined as a number between 0 and 1, computed by statistical number of outcome considered divided by the number of all possible outcomes.

Random Variables

A random variable is a variable whose values subject to varies and assumes to depend on the outcome of an "experiment” (e.g., survey or just observing). A random variable assigns a numerical value to each of outcomes by the number of all outcomes. 

The random variables can be classified as discrete or continuous. The discrete random variables has possible values from a specified list of exact values viz finite or countably infinite. Usually the random variable arises out the counting. On other hand, the continuous random variables can have any numerical value in an interval or collection of intervals. Usually, the continuous random variables arises while measuring certain things. 

The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution.

Expected Value 

The expected value of a random variable is the weighted average of all possible values of random variables with the probability of the value. The expectation of a discrete random variable with possible discrete values x1, x2... together with respective probabilities p1, p2,...

E(X) = ∑ (xi * pi)

Variance of a random variable X is defined as below

 V(X) = E [(X - E(X)) ^ 2] 

 V(X) = E(X^2) - (E(X))^2

Squared root of the variance is the standard deviation of a random variable.

Binomial Distribution

The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n Bernoulli trials. Bernoulli trials are independent experiments with exactly two outcomes viz success and falilure. The probability of success of each trail is p and that of a failure is q = 1 - p in each trial.

Expected value for a random variable with binomial distribution = n * p

Variance for a random variable with binomial distribution = n * p * (1-p)

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

Let X = number of successes in an interval of time or a specific region of space. Then possible values of X are : 0,1,2,… The Poisson probability distribution of X with parameter λ

Normal Distribution
The normal distribution is a continuous probability distribution that has bell-shared probability density function given by following equation:

According to Central Limit Theorem, no matter what a population distribution (relative frequency histogram) is, the sum or average of observations in a random sample taken from the population has approximately normal distribution. This theorem makes the normal distribution to be a most prominently used probability distribution in statistics.