![4.3: Expected Values of
Discrete Random Variables
The variance of a discrete random
variable x is
The standard deviation of a discrete
random variable x is
2 2 2
[( ) ] ( ) ( ).E x x p xσ µ µ= − = −∑
2 2 2
[( ) ] ( ) ( ).E x x p xσ µ µ= − = −∑
10McClave, Statistics, 11th ed. Chapter 4:
Discrete Random Variables](https://image.slidesharecdn.com/chapter04-randomvariablesandprobability-190310074731/85/Chapter-04-random-variables-and-probability-10-320.jpg)
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Chapter 04 random variables and probability
- 1. Chapter 4: Discrete Random Variables Statistics
- 2. McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 2 Where We’ve Been Using probability to make inferences about populations Measuring the reliability of the inferences
- 3. McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 3 Where We’re Going Develop the notion of a random variable Numerical data and discrete random variables Discrete random variables and their probabilities
- 4. 4.1: Two Types of Random Variables A random variable is a variable hat assumes numerical values associated with the random outcome of an experiment, where one (and only one) numerical value is assigned to each sample point. 4McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 5. 4.1: Two Types of Random Variables A discrete random variable can assume a countable number of values. Number of steps to the top of the Eiffel Tower* A continuous random variable can assume any value along a given interval of a number line. The time a tourist stays at the top once s/he gets there *Believe it or not, the answer ranges from 1,652 to 1,789. See Great Buildings 5McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 6. 4.1: Two Types of Random Variables Discrete random variables Number of sales Number of calls Shares of stock People in line Mistakes per page Continuous random variables Length Depth Volume Time Weight 6McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 7. 4.2: Probability Distributions for Discrete Random Variables The probability distribution of a discrete random variable is a graph, table or formula that specifies the probability associated with each possible outcome the random variable can assume. p(x) ≥ 0 for all values of x Σp(x) = 1 7McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 8. 4.2: Probability Distributions for Discrete Random Variables Say a random variable x follows this pattern: p(x) = (.3)(.7)x-1 for x > 0. This table gives the probabilities (rounded to two digits) for x between 1 and 10. x P(x) 1 .30 2 .21 3 .15 4 .11 5 .07 6 .05 7 .04 8 .02 9 .02 10 .01 8McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 9. 4.3: Expected Values of Discrete Random Variables The mean, or expected value, of a discrete random variable is ( ) ( ).E x xp xµ = = ∑ 9McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 10. 4.3: Expected Values of Discrete Random Variables The variance of a discrete random variable x is The standard deviation of a discrete random variable x is 2 2 2 [( ) ] ( ) ( ).E x x p xσ µ µ= − = −∑ 2 2 2 [( ) ] ( ) ( ).E x x p xσ µ µ= − = −∑ 10McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 11. )33( )22( )( σµσµ σµσµ σµσµ +<<− +<<− +<<− xP xP xP Chebyshev’s Rule Empirical Rule ≥ 0 ≅ .68 ≥ .75 ≅ .95 ≥ .89 ≅ 1.00 11McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 4.3: Expected Values of Discrete Random Variables
- 12. 4.3: Expected Values of Discrete Random Variables 12McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables In a roulette wheel in a U.S. casino, a $1 bet on “even” wins $1 if the ball falls on an even number (same for “odd,” or “red,” or “black”). The odds of winning this bet are 47.37% 9986. 0526.5263.1$4737.1$ 5263.)1$( 4737.)1$( = −=⋅−⋅+= = = σ µ loseP winP On average, bettors lose about a nickel for each dollar they put down on a bet like this. (These are the best bets for patrons.)
- 13. 4.4: The Binomial Distribution A Binomial Random Variable n identical trials Two outcomes: Success or Failure P(S) = p; P(F) = q = 1 – p Trials are independent x is the number of Successes in n trials 13McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 14. 4.4: The Binomial Distribution A Binomial Random Variable n identical trials Two outcomes: Success or Failure P(S) = p; P(F) = q = 1 – p Trials are independent x is the number of S’s in n trials Flip a coin 3 times Outcomes are Heads or Tails P(H) = .5; P(F) = 1-.5 = .5 A head on flip i doesn’t change P(H) of flip i + 1 14McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 15. 4.4: The Binomial Distribution Results of 3 flips Probability Combined Summary HHH (p)(p)(p) p3 (1)p3 q0 HHT (p)(p)(q) p2 q HTH (p)(q)(p) p2 q (3)p2 q1 THH (q)(p)(p) p2 q HTT (p)(q)(q) pq2 THT (q)(p)(q) pq2 (3)p1 q2 TTH (q)(q)(p) pq2 TTT (q)(q)(q) q3 (1)p0 q3 15McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 16. 4.4: The Binomial Distribution The Binomial Probability Distribution p = P(S) on a single trial q = 1 – p n = number of trials x = number of successes xnx qp x n xP − =)( 16McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 17. 4.4: The Binomial Distribution The Binomial Probability Distribution xnx qp x n xP − =)( 17McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 18. Say 40% of the class is female. What is the probability that 6 of the first 10 students walking in will be female? 4.4: The Binomial Distribution 1115. )1296)(.004096(.210 )6)(.4(. 6 10 )( 6106 = = = = − −xnx qp x n xP 18McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 19. 4.4: The Binomial Distribution Mean Variance Standard Deviation A Binomial Random Variable has McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 19 2 np npq npq µ σ σ = = =
- 20. 4.4: The Binomial Distribution 16250 2505.5.1000 5005.1000 2 ≅== =⋅⋅== =⋅== npq npq np σ σ µ For 1,000 coin flips, The actual probability of getting exactly 500 heads out of 1000 flips is just over 2.5%, but the probability of getting between 484 and 516 heads (that is, within one standard deviation of the mean) is about 68%. 20McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 21. 4.5: The Poisson Distribution Evaluates the probability of a (usually small) number of occurrences out of many opportunities in a … Period of time Area Volume Weight Distance Other units of measurement 21McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 22. 4.5: The Poisson Distribution ! )( x e xP x λ λ − = = mean number of occurrences in the given unit of time, area, volume, etc. e = 2.71828…. µ = 2 = 22McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 23. 4.5: The Poisson Distribution 1008. !5 3 ! )5( 35 ==== −− e x e xP x λ λ Say in a given stream there are an average of 3 striped trout per 100 yards. What is the probability of seeing 5 striped trout in the next 100 yards, assuming a Poisson distribution? 23McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 24. 4.5: The Poisson Distribution 0141. !5 5.1 ! )5( 5.15 ==== −− e x e xP x λ λ How about in the next 50 yards, assuming a Poisson distribution? Since the distance is only half as long, is only half as large. 24McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables
- 25. 4.6: The Hypergeometric Distribution In the binomial situation, each trial was independent. Drawing cards from a deck and replacing the drawn card each time If the card is not replaced, each trial depends on the previous trial(s). The hypergeometric distribution can be used in this case. McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 25
- 26. 4.6: The Hypergeometric Distribution McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 26 Randomly draw n elements from a set of N elements, without replacement. Assume there are r successes and N-r failures in the N elements. The hypergeometric random variable is the number of successes, x, drawn from the r available in the n selections.
- 27. 4.6: The Hypergeometric Distribution McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 27 − − = n N xn rN x r xP )( where N = the total number of elements r = number of successes in the N elements n = number of elements drawn X = the number of successes in the n elements
- 28. 4.6: The Hypergeometric Distribution McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 28 − − = n N xn rN x r xP )( )1( )()( 2 2 − −− = = NN nNnrNr N nr σ µ
- 29. 4.6: The Hypergeometric Distribution 44.22.2)2()2()2or2( 22. 45 )1)(10( 2 10 22 510 2 5 )2()2( =×==+==== == − − ==== FPMPFMP FPMP Suppose a customer at a pet store wants to buy two hamsters for his daughter, but he wants two males or two females (i.e., he wants only two hamsters in a few months) If there are ten hamsters, five male and five female, what is the probability of drawing two of the same sex? (With hamsters, it’s virtually a random selection.) McClave, Statistics, 11th ed. Chapter 4: Discrete Random Variables 29