Basic Probability Theory and Statistics

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I want to discuss some very fundamental terms /concepts related to probability and statistics.

>Random Experiment:

A Random Experiment is a physical situation whose outcome cannot be predicted until it is observed.

>Sample space :

A Sample Space is a set of all possible outcomes of a random experiment.

Example:

Random Experiment: Toss a fair coin once.

Sample Space: {Head, Tail}

>Random Variables:

A Random Variable is a variable whose possible values are numerical outcomes of a random experiment. There are two types of random variables.

1. Discrete Random Variable is one that may take on only a countable number of distinct values such as 0,1,2,3,4,5,…. Discrete random variables are usually(but not necessarily) counts.

2. Continuous Random Variable is one that takes an infinite number of possible values. Continuous random variables are usually measurements.

>Probability:

Probability is the measure of the likelihood that an event will occur in a Random Experiment. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.

Example:

A simple example is the tossing of a fair(unbiased) coin. Since the coin is fair, the two outcomes(“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of “tails”; is 1/2 (which could also be written as 0.5 or 50%).

>Conditional Probability:

Conditional Probability is a measure of the probability of an event given that (by assumption, presumption, assertion, or evidence) another event has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, is usually written as P(A/B).

>Independence:

Two events are said to be independent of each other if the probability that one event occurs in no way affects the probability of the other event occurring. For Independent events A and B below is true

P(A,B) = P(A)*P(B) where P(A) ≠ 0 and P(B) ≠ 0

P(A/B) = P(A) and P(B/A) = P(A)

Example:

Let’s say you rolled a die and flipped a coin. The probability of getting any number face on the die is in no way influences the probability of getting a head or a tail on the coin.

>Variance:

The variance of a random variable X is a measure of how concentrated the distribution of a random variable X is around its mean. It’s defined as

>Probability Distribution:

A probability distribution is a mathematical function that maps all possible outcomes of a random experiment with its associated probability. It depends on the Random Variable X, whether it’s discrete or continuous.

1. Discrete Probability Distribution:

The mathematical definition of a discrete probability function, P(x), is a function that satisfies the following properties. This is referred to as Probability Mass Function.

2. Continuous Probability Distribution:

The mathematical definition of a continuous probability function, f(x), is a function, f(x), is a function that satisfies the following properties. This is referred to as Probability Density Function.

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