VOCABULARYRANDOM VARIABLE

PROBABILITY DISTRIBUTIONEXPECTED VALUE

LAW OF LARGE NUMBERSBINOMIAL DISTRIBUTION

BINOMIAL RANDOM VARIABLEBINOMIAL COEFFICIENT

GEOMETRIC RANDOM VARIABLEGEOMETRIC DISTRIBUTION

SIMULATION

Random Variables

Key Points

A random variable is a numerical measure(face up number of a die) of the outcomes of a random phenomenon(rolling a die)

If X is a random variable and a and b are fixed numbers, then μₐ₊ᵦₓ= a+βµₓ and Ợ²ₐ₊ᵦₓ=b²Ợ²x

If X and Y are random variables, then μₓ₊ᵧ= μₓ + μᵧIf X and Y are independent random variables, then

Ợ² ₓ₊ᵧ= Ợ²ₓ + Ợ²ᵧ and Ợ² ₓ₋ᵧ= Ợ²ₓ + Ợ²ᵧ As the number of trials in a binomial distribution

gets larger, the binomial distribution gets closer to a normal distribution

Random Phenomenom

Picking a student at random

Random Phenomenom

Clicking a Facebook profile at random

Random Variable

A ______ ______ is a numerical measure of the outcomes of a random phenomenon

The driving force behind many decisions in science, business, and every day life is the question, “What are the chances?”

Picking a student at random is a random phenomenon.

The students grades, height, etc are random variables that describe properties of the student.

Random Variable

The random variables can be: goals inside, goals outside, goals with right foot, etc..

Random Variable

The random variables can be: # of friends, # of miles ran, # of books recently read, etc

Random Variable

The random variables can be categorical as well( top album, movies watched, favorite artists, etc)

Random Variable- Probability distribution

A _______ ________ is a listing or graphing of the probabilities associated with a random variable

Random Variable- Probability(or population) distribution

The probability distribution can be used to answer questions about the variable x( which in this case is the number of tails obtained when a fair coin is tossed three times)

Example: What is probability that there is at least one tails in three tosses of the coin? This question is written as P(X≥1)

P(X≥1)= P(X=1) + P(X=2)+ P(X=3)= 1/8 +3/8+3/8= 7/8

Random variable- discrete and continuous

_______ random variables takes a countable number of values(# of votes a certain candidate receives)

_______ random variables can take all the possible values in a given range(the weight of animals in a certain regions)

Discrete Probability Distribution

Probabilities of certain number of surf boards being sold

Doesn’t make sense for someone to purchase 1.3 surfboards

Continuous Probability Distribution

Infinite values of x are represented with a Continuous Probability Distribution

Random variable- expected value

The mean of the probability distribution is referred to as the ______ ______, and is represented by μₓ.

which just means that the mean(or expected

value) of a random variable is a weighted average

Random Variable- Expected Value

For this probability distribution, the expected value is

= 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8)= 12/8= 1.5

Law of Large Numbers

The _______ of _______ _______states that the actual mean of many trials approaches the true mean of the distribution as the number of trials increases

Rules for Means and Variances of Random Variables

Binomial Distribution

________ ________ models situations with the following conditions:

1. Each observation falls into one of just two categories( success or failure)

2. The number of observations is the fixed number n

3. The n observations are all independent4. The probability of success, p, is the same for

each observation

Binomial Distribution

For data produced with the binomial model, the binomial random variable is the number of successes, X.

The probability distribution of X is a binomial distribution

When finding binomial probabilities, remember that you are finding the probability of obtaining k successes in n trials

Binomial Distribution

Binomial Coefficient

Binomial Distribution

Binomial Coefficient

Binomial Distribution- Calculating Binomial Probability

Binomial Distribution- Calculating binomial probability

Mean and Standard deviation of Binomial Distribution

Geometric Distribution

Each observation falls into one of two categories, success or failure

The variable of interest (usually X) is the number of trials required to obtain the first success

The n observations are all independentThe probability of success, p, is the same for

each observation

Geometric Distribution

Example: If one planned to roll a die until they got a 5, the random variable X= the number of trials until the first 5 occurs.

Find the probability that it would take 8 rolls given that all the conditions of the geometric model are met

Geometric Distribution

If X is a geometric random variable with probability of success P on each trial, then the mean or _______ _______

of the random variable is μ= 1/p.

Expected Value of Geometric Distributions