Probability Distribution - An Introduction

The name itself gives the clear meaning that how the probabilities are distributed over the range of the different values. Let’s make it a bit simple and clear as there are lots of values that a random variable can take and the probabilities of occurrence of those values are different. Say, the chance of coming value ‘x’ within the specified range is 35% and ‘y’ is 45% and ‘z’ is 20%. Clearly, 35% of the probability is distributed to value ‘x’, 45% to ‘y’ & remaining to ‘z’. Thus, how probabilities are distributed over the different random values is shown by the probability distribution.

Why probability distribution?

Suppose you observed people entering into shop some have made a purchase while others not. You wish to know that any particular day 4 people making a successful purchase. Thus, Probability distribution helps in answering such mathematical questions. It helps in finding probabilities, moments (Mean, Variance, skewness). The distribution is useful when you need to know which outcomes are most likely, or their spreads.

What if I say, using probability distribution you can find the probability of death or what is the chance of surviving till a certain age. Even you can use it in speculation on the stock market to check the spread, range of the price and can pocket the handsome gain. Seems interesting?

Let’s go further;

Univariate v/s Multivariate Distribution

If one random variable is used in the distribution, for example, finding X: return on a particular asset is Univariate Probability distribution. They can be discrete or continuous in nature. Whereas,

If more than one variable is used which describes the probabilities of the group of RV is Multivariate probability distribution. For instance, the return of the group of securities, X: return on security A & Y: return on security B. They can be discrete or continuous in nature.

There are various types of univariate and multivariate distribution that are discrete or continuous.

Univariate; Discrete: - Binomial distribution, poison distribution, geometric distribution,

Continuous: - uniform, exponential distribution, gamma distribution, normal dist.

Multivariate; Copula, Gaussian, multivariate normal distribution.

Discrete & continuous distributions?

As probability distribution is based on Random Variable, therefore, distribution can be discrete or continuous.

For discrete RV, a probability distribution defined by a function known Probability mass function, denoted by f(x) or P(X = x). It gives the probability for each value of the variable i.e. point probability. For valid mass function :

f(x) > 0 & sum of the probability of all value equal to 1

Implies function of the distribution is non-negative and takes discrete value.

It is shown as:

For continuous RV, the probability distribution defined by a function known as a Probability density function, denoted by F(x) or P(X <= x). It gives the probability for the range. For valid DF;

F(x) >= 0 & integration of function over the range of ‘x’ equal to 1

Implies function of the distribution is non-negative and takes any real value.

It is shown in the form of function like f(x) = e^x ; where 0<x<1

Lets Check what you have learned

Written by Deepak Agarwal ( pursuing graduation from PGDAV College, University of Delhi and has cleared 4 actuarial papers from IFOA )


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