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Random variables

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  • 1. 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
  • 2. 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
  • 3. Random Phenomenom Picking a student at random
  • 4. Random PhenomenomClicking a Facebook profile at random
  • 5. 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.
  • 6. Random VariableThe random variables can be: goals inside, goals outside, goals with right foot, etc..
  • 7. Random VariableThe random variables can be: # of friends, # of miles ran, # of books recently read, etc
  • 8. Random VariableThe random variables can be categorical as well( top album, movies watched, favorite artists, etc)
  • 9. Random Variable- Probability distribution A _______ ________ is a listing or graphing of the probabilities associated with a random variable
  • 10. 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
  • 11. 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)
  • 12. Discrete Probability DistributionProbabilities of certain number of surf boards being soldDoesn’t make sense for someone to purchase 1.3 surfboards
  • 13. Continuous Probability DistributionInfinite values of x are represented with a Continuous Probability Distribution
  • 14. 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
  • 15. 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
  • 16. 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
  • 17. Rules for Means and Variances of Random Variables
  • 18. 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
  • 19. 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
  • 20. Binomial DistributionBinomial Coefficient
  • 21. Binomial DistributionBinomial Coefficient
  • 22. Binomial Distribution- Calculating Binomial Probability
  • 23. Binomial Distribution- Calculating binomial probability
  • 24. Mean and Standard deviation of Binomial Distribution
  • 25. 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
  • 26. 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
  • 27. 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.