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# Geometric, Poisson, Exponential Dist. CH 15 Week 7 STAT 2332

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This 14 page Class Notes was uploaded by veronica on Tuesday October 11, 2016. The Class Notes belongs to STAT 2332 at University of Texas at Dallas taught by Dr. Chen in Fall 2016. Since its upload, it has received 16 views. For similar materials see Introductory Statistics for Life Sciences in Statistics at University of Texas at Dallas.

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Date Created: 10/11/16

NOTES Week 7 CH 15 (continued): Geometric, Poisson, and Exponential Distributions Quick review: binomial variable X = k = # of successes ????! ???? ????−???? ???? ???? = ???? =) ???? ???? − ???? ) ????! ???? − ???? !) Or P(k) = Now we have geometric distributions Geometric variable X = m = # of trials until 1 success (success happens at m) Ex: Flip a coin until you get Tails, roll a die until you get a 3 Geometric Assumptions/Rules: 1. Each observation falls into success/failure category 2. Observations are independent 3. Probability of “success” p stays constant/the same for each trial 4. X = # of trials it takes for your 1 success Specifically, st X = # of trials it takes for your 1 success m = # of trials where success happens p = probability of success for each trial 1−p = probability of failure for each trial The probability that the first success occurs at mthtrial is ???? ???? = ???? = ???? − ???? ( )????−???? ???? Or P(m) but P(X=m) is more common Example problem: Probability of getting a basketball through the hoop is 0.25. Find the probability that it takes you 5 attempts to successfully make a basketball shot. Solution: P(X = # of attempts it takes for your first successful shot) m = 5 p = 0.25 1-p = 0.75 ????−???? ???? ???? = ???? = ???? − ????.???????? ) ????.???????? ???? ???? ???? = ???? = ????.???????? ( ) ????.???????? = ????.???????????? 6 marbles in a jar. The probability of drawing a red marble is 0.15. (a) Find the probability that you will draw 3 times until you get a red marble. ???? ???? = ???? = ???? − ????.???????? )????−???? ????.???????? (b)Find the probability that you will get a red marble before your 3 draw ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? ( ) st *”before” & “until” mean different things in stats. “Until” is inclusive of the 1 success The probability that at least m trials are needed to get the first success is ????−???? ???? ???? ≥ ???? = ???? − ????( ) Ex: Probability of getting a basketball through the hoop is 0.25. Find the probability that it takes you at least 4 attempts to successfully make a basketball shot. ????−???? ???? ???? ≥ ???? = ???? − ????.???????? ) Find the probability that it takes more than 4 attempts to successfully make a basketball shot. ???? ???? > ???? = ???? − ????.???????? )????−???? Lack of Memory Property: - Information from the past/what happened before doesn’t affect the probability; process resets itself even after consecutive failures Ex: Probability of getting a basketball through the hoop is 0.25. Find the probability that you’ll take at least 5 attempts for you to finally make it in the basket given that you failed the first four times. Solution: “given that” (this is a conditional probability) P(at least 5 more attempts to get 1 successful shot | you failed the first 4 times) P(X ≥ # of attempts it takes for your first successful shot) m = 5 p = 0.25 1-p = 0.75 ???? ???? ≥ ???? = ???? − ????.???????? )????−???? ???? ???? ≥ ???? = ????.???????? ( )???? = ????.???????????? Probability of getting a basketball through the hoop is 0.25. Find the probability that you’ll take 5 more attempts for you to finally make it in the basket given that you failed the first six times. ????−???? ???? ???? = ???? = ???? − ????.???????? ) ????.???????? Quick review: binomial # of successes geometric # of trials Now we have Poisson distributions Poisson # of rare events (represented by X) in time/space/place Ex: number of typing errors per page made by a typist, number of phones exploding in a first-world country A random variable X has a Poisson distribution if the probability that X = k events will occur is given by ???? −???? ???? ???? ???? ???? = ???? = ) ????! ,???? = ????,????,????,????????????. Specifically, P(X = k) = probability that X event will have exactly k instances λ = the average # of events per unit of time or area where different values of λ gives a different Poisson model Ex: The average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see four lions on the next 1-day safari? X = # of lions on the next 1-day safari k = 4 λ = 5 ???? ???? −???? ???? ???? = ???? = ) = ????.???????????? ????! Another example: The average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than three lions on the next 1-day safari? P(X < 3)*remember not to include X = 3 in equation *you can also use P(X ≤ 2) and your k will be 2 X = # of lions on the next 1-day safari k = 3 λ = 5 ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? + ???? ???? = ???? ) ( ) or ???? ???? ≤ ???? = ???? ???? = ???? + ???? ???? = ???? + ???? ???? = ???? ) ( ) Find the probability that tourists will see more than three lions. ???? ???? > ???? = ???? − ????(???? ≤ ????) 0, 1, 2 Relation between Poisson and Binomial When p (success) is small (≤ 0.05) and n is large enough (≥ 20), the number of successes are rare and then Poisson (λ = np) ≈ Binomial(n, p). Poisson distribution ends up as a close approximation of the binomial distribution. Poisson equation (using λ = np for avg number) ≈ Binomial equation Exponential distribution: - Unlike discrete distributions like binomial, geometric, and Poisson, exponential distributions are continuous. - Usually used when talking about time and lifetime Ex: time between marathon runners, lifetime of a car A continuous random variable X is said to follow an exponential distribution if its density curve is given by Where µ = average (expected) value (µ > 0) o Each choice of µ gives a different exponential model. The probability that an exponential variable X will exceed a ???? ???? isvalue −???? /???? ???? ???? > ???? 0) = ???? 0 ????????????????????????????????????????????????????, ???? ???? < ???? 0) = 1 − ???? −???? 0???? *Note that there is no equal sign because ???? ???? = ???? 0) = 0 Ex: Suppose that X = time it takes to buy tickets at a cinema. On average, it takes a 5-minute wait until you get your tickets. Assume that X follows an exponential distribution. - Find the probability that you have to wait for more than 6 minutes. ???? ???? > 6 = ???? −6/5 = 0.301 - Find the probability that you have to wait less than 7 minutes. −7/5 ???? ???? < 7 = 1 − ???? = 0.753 Exponential distributions also have a lack of memory or memory-free property just like geometric distributions. Ex: Lifetime of a functional car follows an exponential model. Given that the car lasted for 10 years, find the probability that the car’s remaining life is 4 years. ???? ???? > 14|???? > 4 = ????(???? > 10) NOTES Week 7 CH 15 (continued): Geometric, Poisson, and Exponential Distributions Quick review: binomial variable X = k = # of successes ????! ???? ????−???? ???? ???? = ???? =) ???? ???? − ???? ) ????! ???? − ???? !) Or P(k) = Now we have geometric distributions Geometric variable X = m = # of trials until 1 success (success happens at m) Ex: Flip a coin until you get Tails, roll a die until you get a 3 Geometric Assumptions/Rules: 1. Each observation falls into success/failure category 2. Observations are independent 3. Probability of “success” p stays constant/the same for each trial 4. X = # of trials it takes for your 1 success Specifically, st X = # of trials it takes for your 1 success m = # of trials where success happens p = probability of success for each trial 1−p = probability of failure for each trial The probability that the first success occurs at mthtrial is ???? ???? = ???? = ???? − ???? ( )????−???? ???? Or P(m) but P(X=m) is more common Example problem: Probability of getting a basketball through the hoop is 0.25. Find the probability that it takes you 5 attempts to successfully make a basketball shot. Solution: P(X = # of attempts it takes for your first successful shot) m = 5 p = 0.25 1-p = 0.75 ????−???? ???? ???? = ???? = ???? − ????.???????? ) ????.???????? ???? ???? ???? = ???? = ????.???????? ( ) ????.???????? = ????.???????????? 6 marbles in a jar. The probability of drawing a red marble is 0.15. (a) Find the probability that you will draw 3 times until you get a red marble. ???? ???? = ???? = ???? − ????.???????? )????−???? ????.???????? (b)Find the probability that you will get a red marble before your 3 draw ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? ( ) st *”before” & “until” mean different things in stats. “Until” is inclusive of the 1 success The probability that at least m trials are needed to get the first success is ????−???? ???? ???? ≥ ???? = ???? − ????( ) Ex: Probability of getting a basketball through the hoop is 0.25. Find the probability that it takes you at least 4 attempts to successfully make a basketball shot. ????−???? ???? ???? ≥ ???? = ???? − ????.???????? ) Find the probability that it takes more than 4 attempts to successfully make a basketball shot. ???? ???? > ???? = ???? − ????.???????? )????−???? Lack of Memory Property: - Information from the past/what happened before doesn’t affect the probability; process resets itself even after consecutive failures Ex: Probability of getting a basketball through the hoop is 0.25. Find the probability that you’ll take at least 5 attempts for you to finally make it in the basket given that you failed the first four times. Solution: “given that” (this is a conditional probability) P(at least 5 more attempts to get 1 successful shot | you failed the first 4 times) P(X ≥ # of attempts it takes for your first successful shot) m = 5 p = 0.25 1-p = 0.75 ???? ???? ≥ ???? = ???? − ????.???????? )????−???? ???? ???? ≥ ???? = ????.???????? ( )???? = ????.???????????? Probability of getting a basketball through the hoop is 0.25. Find the probability that you’ll take 5 more attempts for you to finally make it in the basket given that you failed the first six times. ????−???? ???? ???? = ???? = ???? − ????.???????? ) ????.???????? Quick review: binomial # of successes geometric # of trials Now we have Poisson distributions Poisson # of rare events (represented by X) in time/space/place Ex: number of typing errors per page made by a typist, number of phones exploding in a first-world country A random variable X has a Poisson distribution if the probability that X = k events will occur is given by ???? −???? ???? ???? ???? ???? = ???? = ) ????! ,???? = ????,????,????,????????????. Specifically, P(X = k) = probability that X event will have exactly k instances λ = the average # of events per unit of time or area where different values of λ gives a different Poisson model Ex: The average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see four lions on the next 1-day safari? X = # of lions on the next 1-day safari k = 4 λ = 5 ???? ???? −???? ???? ???? = ???? = ) = ????.???????????? ????! Another example: The average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than three lions on the next 1-day safari? P(X < 3)*remember not to include X = 3 in equation *you can also use P(X ≤ 2) and your k will be 2 X = # of lions on the next 1-day safari k = 3 λ = 5 ???? ???? < ???? = ???? ???? = ???? + ???? ???? = ???? + ???? ???? = ???? ) ( ) or ???? ???? ≤ ???? = ???? ???? = ???? + ???? ???? = ???? + ???? ???? = ???? ) ( ) Find the probability that tourists will see more than three lions. ???? ???? > ???? = ???? − ????(???? ≤ ????) 0, 1, 2 Relation between Poisson and Binomial When p (success) is small (≤ 0.05) and n is large enough (≥ 20), the number of successes are rare and then Poisson (λ = np) ≈ Binomial(n, p). Poisson distribution ends up as a close approximation of the binomial distribution. Poisson equation (using λ = np for avg number) ≈ Binomial equation Exponential distribution: - Unlike discrete distributions like binomial, geometric, and Poisson, exponential distributions are continuous. - Usually used when talking about time and lifetime Ex: time between marathon runners, lifetime of a car A continuous random variable X is said to follow an exponential distribution if its density curve is given by Where µ = average (expected) value (µ > 0) o Each choice of µ gives a different exponential model. The probability that an exponential variable X will exceed a ???? ???? isvalue −???? /???? ???? ???? > ???? 0) = ???? 0 ????????????????????????????????????????????????????, ???? ???? < ???? 0) = 1 − ???? −???? 0???? *Note that there is no equal sign because ???? ???? = ???? 0) = 0 Ex: Suppose that X = time it takes to buy tickets at a cinema. On average, it takes a 5-minute wait until you get your tickets. Assume that X follows an exponential distribution. - Find the probability that you have to wait for more than 6 minutes. ???? ???? > 6 = ???? −6/5 = 0.301 - Find the probability that you have to wait less than 7 minutes. −7/5 ???? ???? < 7 = 1 − ???? = 0.753 Exponential distributions also have a lack of memory or memory-free property just like geometric distributions. Ex: Lifetime of a functional car follows an exponential model. Given that the car lasted for 10 years, find the probability that the car’s remaining life is 4 years. ???? ???? > 14|???? > 4 = ????(???? > 10)

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