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Probability Distributions

Probability Distributions. Chapter 7. The random variable X has a single value for each outcome within an experiment Discrete variables have separate and equi -distant values from each other Ex. 1, 2, 3, and 4 Continuous variables have an infinite number of possible values Ex. 1.2367

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Probability Distributions

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  1. Probability Distributions Chapter 7

  2. The random variable X has a single value for each outcome within an experiment • Discrete variables have separate and equi-distant values from each other Ex. 1, 2, 3, and 4 • Continuous variables have an infinite number of possible values Ex. 1.2367 • The probability that X will take on a particular value is P(x) • If all the values are equally likely than you have a discrete uniform distribution • The formula is P(x) = 1/n • An expected value is the average of all possible outcomes of a probability experiment E(x) = x1 P(x1) + x2 P(x2) … + xnP(xn)

  3. Suppose you had eight 5cm straws, seven 3cm straws, and three 6cm straws • Show the probability distribution for the length of a given straw • Determine the expected length of a straw E(x) = x1P(x1) + x2 P(x2) …+xn P(xn) = (5)(8/18) + (3)(7/18) + (6)(3/18) = 2.2 + 1.2 + 1 = 4.4

  4. A binomial distribution has a specified number of independent trails in which the outcome is either success of failure • The probability of success is the same in each trail • The expectation for binomial distribution is E(x)= np • The probability is P(x) = (nCx)(p^x)(q^n-x)

  5. Example • 20% of the population is right handed • What is the probability that 4/10 people are right handed? • What is the expected number of right handed people? • P (success) =0.2 • Q (failure) =0.8 • P(x) = (10C4)(0.2^4)(0.8^6) • =0.09 • E(x) =np • =(10)(0.2) • =2

  6. A geometric distribution has a specified number of independent trails with two possible outcomes success or failure • It is either a success or failure once again, however you are counting the waiting time • The probability is P(x) = (q^x)(p) • The expectation of a geometric distribution is E(X) = q/p

  7. Example • You get a cake for an A- average • The probability of getting an A- on any one of the tests is 10% • What is the probability of getting the pizza on the 4th test? • P =0.1 • Q = 1 - 0.1 • = 0.9

  8. 7.4 Hypergeometric Distributions • A hypergeometric distribution has a specified number of dependant trails having the only possible outcomes success or failure • Probabilities are dependent and are not equal • The individual outcomes be repeated within these trails • P(x)= (aCx)(n-aCr-x) /nCr • The expectation for hypergoemetric distribution is E(x)= (r)(a)/n

  9. Example • Of 100 students, 50 are taking math, 5 are randomly selected • What is the probability that 3 are taking math?

  10. Activity • For each correct answer, you will be awarded a letter • Once you collect enough letters you have to unscramble them to make a word • The first group to make the word wins

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