If we assign numbers to the outcomes — say, 1 for heads, 0 for tails — then we have created the mathematical object known as a random variable. The expectation of a random variable is a number that attempts to capture the center of that random variable's distribution. It can be interpreted as the long-run average of many independent samples from the given distribution. More precisely, it is defined as the probability-weighted sum of all possible values in the random variable's support,.

## Probability and measure pdf

Consider the probabilistic experiment of rolling a fair die and watch as the running sample mean converges to the expectation of 3. Change the distribution of the different faces of the die thus making the die biased or "unfair" by adjusting the blue bars below and observe how this changes the expectation. Whereas expectation provides a measure of centrality, the variance of a random variable quantifies the spread of that random variable's distribution.

The variance is the average value of the squared difference between the random variable and its expectation,. Draw cards randomly from a deck of ten cards.

As you continue drawing cards, observe that the running average of squared differences in green begins to resemble the true variance in blue. Chapter 1 Basic Probability This chapter is an introduction to the basic concepts of probability theory.

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Chance Events Randomness is all around us. Flip the Coin. Consider our roll a die example again. On a single die roll, 1 and 2 and 3 and 4 and 5 and 6 are mutually exclusive because one and only one number from the sample space can appear face up.

Events are collectively exhaustive if they constitute the entire set of possibilities from the sample space, and no other outcomes are possible. Events are independent if the occurrence of one does not affect the occurrence of any other subsequent event.

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## Basic Probability Theory (3)

If we roll a die twice, it results in independent events as in the first roll getting a specific number face up does not in any manner affects which number will come face up on the second roll. Events are dependent when occurrence of one event affects the occurrence of any other subsequent event. For example, if have a deck of cards and we draw 2 cards one after another and compute probability of getting 2 clubs, without replacing the card drawn first, it will make the second event dependent on the first.

Let us say there are A and B independent events then the possibility of A and B occurring together is the product of their individual probabilities:. The subtraction of P AB is to avoid double counting.

### Your Answer

To get a better understanding, consider an example - drawing a jack or a club from a deck of cards. In this experiment, we can draw either a club or a jack or a jack of club. Note : If A precludes B or vice versa, then clearly P AB is equal to "0" converting the above equation into a normal addition rule for mutually exclusive events. Another way to build better understanding is to compute the probability of not drawing a jack or a club and then subtract it from 1 to arrive at the desired probability:.

To understand the concept let us begin with questions. What is the probability of a specific sequence of four coin tosses, "heads-tails-heads-heads"? And what is the probability of observing 3 "heads" and 1 "tails" in any order? Whereas in the second case there are 4 possible ways in which we can get 3 "head" and 1 "tails" i. To understand this, let us take an example - how many different sequences can we create from letters "A", "B", "C", and "D"?

The first letter of a sequence can be any one of the four.

## Basic Probability Theory

After drawing one, the second letter of the sequence can be any of the remaining three letters. The third letter can be any of the remaining two letters, and the fourth must be the one remaining letter.

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