Quick Intro To Probability Distributions:

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails (assuming the coin is fair)

Two Types of Random Variables:

Discrete: X is discrete if it takes a few values(mass-points) x1, x2, …xn.. with corresponding probabilities p1, p2..pn. The probability law specifying the probabilities for different values is called probability mass function f(X) = P(X=x), for x – x1, x2…xn and it satisfies the following two conditions:

• f(x) ≥ 0 
• Σ f(x) = 1, sum being taken over all values of x 

Mean = μ = E(X) = Expected Value of X = Σ x.f(x) = Σ (value . probability) SD = σ = √ Variance, where Variance = E(X - μ )2 = E(X2) - μ2 = Σ x2.f(x) - μ2. 

While mean tells the individual values the SD σ indicates how concentrated individual values of the distribution are with respect to the mean

Chebyshev’s Inequality: For any distribution with mean m and SD s, the interval (m - ks, m + ks) covers at least (1-1/k2) proportion of the total values.

For Symmetric bell-shaped distribution (Normal distribution), the interval (m - s, m + s) covers approximately 68%, the interval (m - 2s, m + 2s) covers approximately 95%, and (m - 3s, m + 3s) covers approximately 99.73%, proportions.

Some Discrete distributions:

Binomial Distribution:

The distribution is applicable to following type of experiments:

•  Only two outcomes, success(S) or failure(F)
• P(S) = p and P(F) = 1- p = q
• They are probabilistically independent

If total no of trials are n and suppose X= number of successes out of n trials. Then probability function:

f(x) = P(X =x) = nCx.pxqn-x for x = 0,1, ..., n = 0, otherwise.   

The mean and SD for this distribution are : Mean = m = np and SD = s = Önpq 

Poisson distribution:

Unlike binomial distribution the random variable X can take any positive integer value including zero. The probability mass function is

f(x) = P(X = x) = e-m.mx/x! , for x = 0, 1, 2, ..., ∞, = 0, otherwise. 

For Poisson distribution, Mean = m and SD = Öm

The Poisson distribution is always positively skewed. This model is good for fitting discrete data which are rare to occur.

Continuous: A random variable is X continuous if it takes any values in an interval (a, b). Since the number of values in (a,b) is uncountable, we cannot assign probabilities to distinct values of the variable, instead we consider a continuous function f(x) over the interval (a.b) such that probability for X to take values in any sub interval is obtained as the area bounded by the function over the sub-interval. Such a function is called probability density function(p.d.f). It has following two conditions:

• f(x) ≥ 0 for all x
• Total area under f(x) between a and b = 1. 

Normal distribution: The most widely continuous distribution is normal distribution.it is perfectly symmetrical distribution with mean μ and SD s and defined over entire real line (- ∞, ∞). Some of the properties of this distribution are:

• Distribution is symmetrical about its mean m, so area below m = ½ = area above m
• Mean = Median = Model = m
• SD s is such that the curve changes its curvature at m-s and m+s
• The standard variable Z = (X-m)/s is also normal with mean = 0 and SD = 1.

Joint probability distribution of two random variables:

Consider two discrete random variables X and Y having n1 values (x1, x2, .. xn1) and Y having n2 values (y1, y2, …yn2). The joint probability distribution between X and Y is given by

Pij = P( X = xi, Y = yj), for i = 1, 2, ..., k, and j = 1, 2, ..., l. 

Naturally, we must have Pij ≥ 0 for all (i, j) and ∑Pij = 1.

A measure of dependence – Correlation coefficient between two random variables:

It is a measure of linear relationship that exist between two variables. This is defined as

Sign of ρ tells how one variable behaves with variation in other:

• If both behaves in same direction (both increases or both decreases) then it is positive correlation 0 < ρ <1
•  If both behaves in opposite direction (one increases as other decreases and conversely) then it is negative correlation -1 < ρ <0
• Case when ρ = 0: Here the two variables are called uncorrelated. This means that there is no linear relationship between the two variables.