Random Variables VOCABULARY RANDOM VARIABLE PROBABILITY DISTRIBUTION EXPECTED VALUE LAW OF LARGE NUMBERS BINOMIAL DISTRIBUTION BINOMIAL RANDOM VARIABLE BINOMIAL COEFFICIENTGEOMETRIC RANDOM VARIABLE GEOMETRIC DISTRIBUTION SIMULATION
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 + μᵧ If 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 PhenomenomClicking 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 VariableThe random variables can be: goals inside, goals outside, goals with right foot, etc..
Random VariableThe random variables can be: # of friends, # of miles ran, # of books recently read, etc
Random VariableThe 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 DistributionProbabilities of certain number of surf boards being soldDoesn’t make sense for someone to purchase 1.3 surfboards
Continuous Probability DistributionInfinite 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 ValueFor this probability distribution, theexpected 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 n3. 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- 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 independent The probability of success, p, is the same for each observation
Geometric DistributionExample: If one planned to roll a die until they got a 5, the randomvariable X= the number of trials until the first 5 occurs.Find the probability that it would take 8 rolls given that all theconditions of the geometric model are met
Geometric Distribution Expected Value of Geometric DistributionsIf X is a geometric random variable with probability of success Pon each trial, then the mean or _______ _______ of therandom variable is μ= 1/p.