# 6.1: Random Variables

## Random Variables

• Random variable: a variable whose value is a numeric outcome of a random event

• Not 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 probabilities

• To 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

• Continuous random variable: a variable with an “uncountable” number of individual outcomes

• With 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

## Big Ideas

### Big Idea #1: Adding two random variables

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

### Big Idea #2: Subtracting two random variables

• New mean — subtract the two original means

• New variance — add the two original variances

• New standard deviation — square root the new variance

### Big Idea #3: Multiplying the random variable by a constant

• 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

### Big Idea #4: Adding a constant, C, to a random variable

• New mean — add C to the original mean

• New variance — same as original variance

• New standard deviation — same as original standard deviation