Random Variables

Random variable

Let be the sample space of an experiment. A function , which assigns a real number to every is called a random variable.

Range space

Each possible value of corresponds to an event that is a subset or element of the sample space .

Notation

• upper case letters to denote random variables
• lower case letters to denote observed values

Probability Distribution

For this course, there are two main types of random variables.

1. Discrete random variables (number of values in is finite)
2. Continuous random variable (number of values in is infinite)

Discrete random variables

Probability mass function for discrete variables

For a discrete random variable , define the probability mass function

Properties:

1. for all
2. for all
3. or

Continuous random variables

Probability density function for continuous variables

For a continuous random variable , is a probability density function that satisfies

1. Non-negativity for all , for
2. Sum of all probabilities add up to 1 . This particular condition can be represented as
3. For any where

The probability density function describes the shape of the distribution.

Checking probability density function

It suffices to check conditions 1 and 2.

1. for all , for

Note that for any specific value of , we have

This is an example of: , but is not necessarily .

Inequalities

As represents the area under the graph, it can be represented as

4. )

Cumulative Distribution Function

Cumulative distribution function

For any random variable , the cumulative distribution function is defined by

Definition is applicable regardless of type of random variable (discrete or continuous)

Discrete random variables

If is a discrete random variable,

For any two numbers ,

where represents the largest value in smaller than :

Continuous random variable

If is a continuous random variable,

and

Further,

1. Regardless of type, is non-decreasing.

2. Probability function and cumulative distribution are one-to-one correspondence. That is, for any probability function given, the cumulative distribution function is uniquely determined.
3. The ranges of and satisfy:
   2. for discrete distributions,
   3. for continuous, but not necessarily .

Properties

Right-continuous Cumulative distribution function is continuous except possibly for having some jumps. When there is a jump, the cumulative distribution function is continuous from the right.

Convergence to 0 and 1 in limits

Title

Mention

Probability Distribution

For this course, there are two main types of random variables.

1. Discrete random variables (number of values in is finite)
2. Continuous random variable (number of values in is infinite)

Discrete random variables

Probability mass function for discrete variables

For a discrete random variable , define the probability mass function

Properties:

1. for all
2. for all
3. or

Continuous random variables

Probability density function for continuous variables

For a continuous random variable , is a probability density function that satisfies

1. for all , for
3. For any where

Checking probability density function

It suffices to check conditions 1 and 2.

1. for all , for

Note that for any specific value of , we have

This is an example of: , but is not necessarily .

Cumulative Distribution Function

Cumulative distribution function

For any random variable , the cumulative distribution function is defined by

Remark

Definition is applicable regardless of type of random variable (discrete or continuous)

Discrete random variables

If is a discrete random variable,

For any two numbers ,

where represents the largest value in smaller than :

Continuous random variable

If is a continuous random variable,

and

Further,

1. Regardless of type, is non-decreasing.

2. Probability function and cumulative distribution are one-to-one correspondence. That is, for any probability function given, the cumulative distribution function is uniquely determined.
3. The ranges of and satisfy:
   2. for discrete distributions,
   3. for continuous, but not necessarily .

Properties

Right-continuous Cumulative distribution function is continuous except possibly for having some jumps. When there is a jump, the cumulative distribution function is continuous from the right.

Convergence to 0 and 1 in limits

Expectation

The expectation of a random variable (denoted either or )is the average value of it if the corresponding experiment is repeated many times.

Expectation: Discrete random variable

Let be a discrete random variable with and a probability function . The expectation or mean is then defined:

Expectation: Discrete random variable

Let be a discrete random variable with and a probability function . The expectation or mean is then defined:

Expectation: Continuous random variable

Let be a continuous random variable with a probability function . The expectation or mean is then defined:

Note that the mean of is not necessarily a possible value of the random variable :

• the expectation of rolling a dice may not be any of the values {1, 2, 3, 4, 5, 6}

Properties

1. Let be a random variable, and let be any real numbers ().

2. Let be two random variables. We have:

3. Let be an arbitrary function.

• if is a discrete random variable with probability mass function & range ,

• if is a continuous random variable with probability density function & range ,

We can then derive the following from properties (1) and (2): for constants , and random variables :

Variance

Variance

The variance of a random variable is defined:

An alternative formula also can be used:

This definition applies regardless of the type of variable (whether is discrete, or continuous).

If is a discrete random variable with probability mass function and range :

If is a continuous random variable with probability density function :

The variance of a variable is non-negative:

Equality holds if & only , that is when is a constant.

Let be any real numbers, then:

The positive square root of the variance is defined as the standard deviation of :