## Probability Distribution

The probability distribution of a random variable “X” is basically a graphical presentation of the probabilities of all possible outcomes of X. A random variable is any quantity for which more than one value is possible, for instance, the price of quoted stocks. Simply put, a probability distribution gathers all the outcomes and goes a step further to indicate the probability associated with each outcome.

## Discrete Random Variables

A discrete random variable is one that can take on a finite number of outcomes.

Examples:

1. If we roll a dice, there are 6 possible outcomes. Therefore, the outcomes are discrete and random.
2. The number of CFA finalists employed within a given year is a discrete random variable.

## Continuous Random Variables

A continuous random variable is that which has an infinite number of possible outcomes. A good example can be the rate of return on a stock. For instance, the return can be 6% or between 6% and 7%, in which case it can take on 6.4%, 6.41%, 6.412% or even 6.412325%, i.e., infinite values.

## Probability Function

A probability function gives the probability of a random variable X taking on a value “x”. The probability functions of discrete and continuous random variables are slightly different:

For a discrete random variable, the probability function, P(x), satisfies the following properties:

1. P(X = x) = P(x)
In statistics, P(x) is said to be a probability mass function
2. P(x) is always nonnegative for all x
3. The sum of the probabilities of all possible outcomes = 1.

For a continuous random variable, the probability function, f(x), satisfies the following properties:

1. The probability that x is between two values, a and b is
$$P\left\{ a\le x\le b \right\} =\int _{ a }^{ b }{ f\left( x \right) dx }$$
f(x) is said to be a probability density function(pdf)
2. f(x) is nonnegative for all x
3. The integral of the probability function is 1, that is
$$\int _{ -\infty }^{ \infty }{ f\left( x \right) dx=1 }$$

Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.

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