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CHAPTER 6: DISCRETE PROBABILITY DISTRIBUTIONS

CHAPTER 6: DISCRETE PROBABILITY DISTRIBUTIONS. PROBIBILITY DISTRIBUTION DEFINITIONS (6.1):. Random Variable is a measurable or countable outcome of a probability experiment. Discrete Random Variables have countable and finite outcomes.

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CHAPTER 6: DISCRETE PROBABILITY DISTRIBUTIONS

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  1. CHAPTER 6:DISCRETE PROBABILITY DISTRIBUTIONS

  2. PROBIBILITY DISTRIBUTION DEFINITIONS (6.1): • Random Variable is a measurable or countable outcome of a probability experiment. • Discrete Random Variables have countable and finite outcomes. • Continuous Random Variables have infinitely many possible outcomes from measurements and are described in ranges of values.

  3. DISCRETE PROBABILITY DISTRIBUTIONS: • Looks like Discrete Relative Frequency Distributions from chapter 2. • The relative frequency is the probability [p(x)] of the discrete value (x) occurring and therefore has a value between 0 and 1. • The sum of all probabilities is 1.0.

  4. DISCRETE PROBABILITY DISTRIBUTIONS: • Example of rolling a pair of dice. Will do this together.

  5. DISCRETE PROBABILITY DISTRIBUTIONS: • Make up of Jury in area that is 65% Hispanic.

  6. DISCRETE PROBABILITY DISTRIBUTIONS: • Mean of the distribution: • Std. Dev. of the Distribution: • Calculator:

  7. EXPECTED VALUE: • Same as discrete probability distribution but some outcomes may be negative. • Calculated same as mean of discrete probability distribution: • Example of raffle. • Example of lottery. • Example of insurance.

  8. BINOMIAL PROBABILITY DISTRIBUTION (6.2): • Calculate the probability of rolling a die five times and getting a “1” exactly 2 times. • p=1/6, n=5, 1-p=5/6. • How many ways can this happen. • Do the same for 0, 1, 3, 4 and 5 times.

  9. BINOMIAL PROBABILITY DISTRIBUTION: • Let • n = number of trials • p = probability of success on any given trial • q = 1 – p = probability of failure on any given trial • x = the desired number of successes out of n trials • Then probability of x successes from n trials

  10. BINOMIAL PROBABILITY DISTRIBUTION: • REQUIREMENTS FOR AN EXPERIMENT TO BE A BINOMIAL PROBABITY PROCEDURE: • Fixed number of trials (n); • Fixed probability (p) of success for every trial; • Each trial independent of all other trials; • Each trial has only TWO possible outcomes.

  11. BINOMIAL PROBABILITY DISTRIBUTION: • Example: n=8, p=0.85, find p(5). • Use formula • Use table • Use calculator: • DISTR (2nd, VARS)0.binompdf(n,p,x) • Do also for x = 0, 1, 2, 3, 4, 6, 7, 8 and build discrete probability distribution; graph result.

  12. BINOMIAL PROBABILITY DISTRIBUTION: • Do again with n=8, p=0.22 • Build discrete probability distribution and graph. • Compare graphs. • Skew is result of mean (n*p) • As n increases the graph becomes more bell shaped.

  13. BINOMIAL PROBABILITY DISTRIBUTION: • Mean of Binomial Distribution: • Std. Dev. Of Binomial Distribution: • Example

  14. BINOMIAL PROBABILITY DISTRIBUTION: • Cumulative Probability: • Findp(x ≤ 2), p(x < 2), p(x ≥ 2), p(x > 2) • Use Calculator (always gives p(x ≤ c), DISTR (2nd, VARS)A.binomcdf(n,p,x)

  15. BINOMIAL PROBABILITY DISTRIBUTION: • Cumulative Probability: • Use Calculator:(always gives p(x ≤ c) • For p(x ≤ c) binomcdf(n,p,x) • For p(x < c) binomcdf(n,p,x-1) • For p(x ≥ c) 1 - binomcdf(n,p,x-1) • For p(x > c) 1 - binomcdf(n,p,x) USE NUMBER LINE TO FIND X

  16. BINOMIAL PROBABILITY DISTRIBUTION: • 65% of population of city is Hispanic. A jury of 12 is selected. • Is this Binomial? Why or why not? • What is n, and p? • Find p(x=7); p(X=2); p(x≤6); p(x>6). • Would it be unusual to find a jury with only 2 Hispanics. • Find the mean and std. dev. of this distribution • Is it unusual to have less than 7 Hispanic jurors; how about 4?

  17. POISSON DISTRIBUTION: (6.3) • Defined as the countable number of events in a fixed interval. • The number of defects in a square yard of material. • The number of Okapi’s in a square mile of rain forest. • The number speeding tickets issued in a day in a city. • Look at 8 x 11 inch paper with dots.

  18. POISSON DISTRIBUTION: • REQUIREMENTS FOR X TO BE A POISSON RANDOM VARIABLE: • The probability for two or more events in a sufficiently small interval is 0. • The probability is the same for any two equal intervals. • The number of successes in any interval is independent of the number of successes in any other interval, providing the intervals are not overlapping

  19. POISSON DISTRIBUTION: • Let λ be the mean number of events per unit interval and t be the number of intervals. • Then the mean number of events in t intervals is t* λ = µ. • Then • Mean of the Poisson Distribution = µ • Std. Dev. of the Poisson Distribution = • Calculator: DISTR (2nd, VARS)B.poissonpdf(µ,x)

  20. POISSON DISTRIBUTION: • In the Binomial Distribution the minimum value for x is 0 and the maximum is n. • In the Poisson distribution the minimum value of x is 0, but the maximum value is infinity.

  21. POISSON DISTRIBUTION: • Example: 3 defects per sq. yd.; let n = 10 sq. yds.; Find P(x = 20) • Example: 1 ticket per day per mile; n = 4 miles; Find P(x = 3); also P(x ≤ 3). • Example: 6 Okapis per Sq. mile; let n = 10 sq. miles; Find P(x = 70); also P(x ≥ 70)

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