6.1: Random Variables

**Random variable**: a variable whose value is a numeric outcome of a random eventNot a random variable — the outcome of heads or tails in a coin flip

Random variable — counting the number of tails in 5 flips

**Discrete random variable x**: has a countable number of possible values**Probability distribution of x**: lists the values X can have and their corresponding probabilitiesTo be a legitimate probability distribution,

Each probability must be between 0 and 1

The sum of the probabilities must be 1

The mean is also called the expected value because in the long run of many trials, that is the value one would expect to get on average

Interpreting the mean (expected value)

If we repeat [trial] many times, we expect to get [average] on average per [trial].

Interpreting the standard deviation

On average, the number of [value] will differ from the mean by [standard deviation].

**Continuous random variable**: a variable with an “uncountable” number of individual outcomesWith CRVs, it makes no sense to talk about individual outcomes

Instead, we talk about a range of outcomes using areas under a density curve

For CRVs, we can’t set up probability distributions

A CRV x takes all the values in an interval of numbers

**Interval**: a set of real numbers that contains all real numbers lying between any two numbers of the set

The probability distribution of x is the area under the curve for the interval that x takes

**Normal distribution**: can be used for a continuous random variable probability distribution because the area under a normal curve is equal to 1

For variance and standard deviation, the variables must be independent. If we are not told that they are, we can only add the means (we will generally be given that they are independent.

New mean — add the two original means

New variance — add the two original variances

New standard deviation — square root the new variance

New mean — subtract the two original means

New variance — add the two original variances

New standard deviation — square root the new variance

New mean — multiply the original mean by b

New variance — multiply the original variance by b^2

New standard deviation — square root the new variance

New mean — add C to the original mean

New variance — same as original variance

New standard deviation — same as original standard deviation

**Random variable**: a variable whose value is a numeric outcome of a random eventNot a random variable — the outcome of heads or tails in a coin flip

Random variable — counting the number of tails in 5 flips

**Discrete random variable x**: has a countable number of possible values**Probability distribution of x**: lists the values X can have and their corresponding probabilitiesTo be a legitimate probability distribution,

Each probability must be between 0 and 1

The sum of the probabilities must be 1

The mean is also called the expected value because in the long run of many trials, that is the value one would expect to get on average

Interpreting the mean (expected value)

If we repeat [trial] many times, we expect to get [average] on average per [trial].

Interpreting the standard deviation

On average, the number of [value] will differ from the mean by [standard deviation].

**Continuous random variable**: a variable with an “uncountable” number of individual outcomesWith CRVs, it makes no sense to talk about individual outcomes

Instead, we talk about a range of outcomes using areas under a density curve

For CRVs, we can’t set up probability distributions

A CRV x takes all the values in an interval of numbers

**Interval**: a set of real numbers that contains all real numbers lying between any two numbers of the set

The probability distribution of x is the area under the curve for the interval that x takes

**Normal distribution**: can be used for a continuous random variable probability distribution because the area under a normal curve is equal to 1

For variance and standard deviation, the variables must be independent. If we are not told that they are, we can only add the means (we will generally be given that they are independent.

New mean — add the two original means

New variance — add the two original variances

New standard deviation — square root the new variance

New mean — subtract the two original means

New variance — add the two original variances

New standard deviation — square root the new variance

New mean — multiply the original mean by b

New variance — multiply the original variance by b^2

New standard deviation — square root the new variance

New mean — add C to the original mean

New variance — same as original variance

New standard deviation — same as original standard deviation