Chapter 5 Intro to Random Variables

Random Variable

Takes numerical values that describe the outcomes of a chance process

What are the two main types of Random Variables?

Discrete and Continuous

Discrete Random Variables

Takes a set of individual values with gaps between them on a number line, where the possible values can be finite or infinite

Continuous Random Variable

Takes any value in a particular interval on the number line, where the set of possible values is infinite

Discrete Random Variables result from _____ something. Continuous Random Variables result from _____ something

Counting; Measuring

Probability Distribution

Gives a random variable its possible values and their probabilities

What are the 2 points needed for a Probability Distribution to be valid?

1. Every probability is a number between 0 and 1

2. The sum of the probabilities is 1

How to label the Mean of a Random Variable?

μ_x

Expected Value (mean) of a Discrete Random Variable

Its average; to find, multiply each possible value of X by its probability and then add

E(X) = Σ x₁p₁+x₂p₂+…+x_ip_i

How to label the Standard Deviation of a Probability Distribution?

σ_x

Standard Deviation of a Discrete Random Variable

Measures how much the values of the variable typically vary from the mean

σ_x = sqrt( Σ(x_i-μ_x)²p_i )

Binomial Setting

Arises when we perform “n” independent trials of the same chance process and count the number of times a particular outcome (called a “success”) occurs

What are the 4 conditions for a Binomial Setting?

1. Binary - outcomes can be classified as “success” or “failure”

2. Independent - knowing the result of one trial doesn’t say anything about another trial

3. Number - the number of trials “n” is fixed in advance

4. Same Probability - there is the same probability of success “p” on each trial

Binomial Probability Formula

P(X=r) = _nC_r(p)^r(1-p)^n-r where _nC_r = n!/r!(n-r)!

Binomial Distribution

The probability distribution of a binomial random variable that is completely satisfied by 2 numbers:

1) the number of trials “n”, and

2) the probability of success on each trial “p”

How to find the Mean of a Binomial Random Variable?

μ_x = np

How to find the Standard Deviation of a Binomial Random Variable?

σ_x = sqrt(np(1-p))