A random variable .
Write down the integer values of and which make the following statements true:
Use calculations of the form with to find:
Give your answers to three significant figures.
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A random variable .
Write down the integer values of and which make the following statements true:
Use calculations of the form with to find:
Give your answers to three significant figures.
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A random variable .
Use calculations of the form to find exact values, in terms of , for
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The random variable.
Find the value of in the case when:
Find the value of in the case when:
Given that , find a simplified expression in terms of for:
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For each of the following random variables, calculate the mean using and the standard deviation using .
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The random variable .
Calculate:
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A random variable .
Find the exact value, in terms of , of:
Use your answers to part (a) to find the exact value of:
Compare your answers to part (a) and part (b), then use the pattern to write down an expression, in terms of , for
where is an integer and .
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Naomi, a teacher, receives emails at a constant average rate during the school day. Naomi models the number of emails she receives within a fixed period of time using a Poisson distribution, .
Write down two assumptions that Naomi has made about the emails in order to use a Poisson model.
Naomi knows that on average she receives 10 emails in one hour.
Calculate the value of in the case when Naomi is modelling the number of emails she receives in:
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Use the tables of values for the Poisson cumulative distribution function to calculate the probabilities in this question.
The random variable , find:
The random variable , find:
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Amira has a bad Internet connection at her house. Her internet disconnects on average 5 times each day.
Define a suitable distribution to model the number of times the internet at Amira’s house disconnects during a day. State any assumptions you make.
Find the probability that during a random day the internet at Amira’s house disconnects:
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Lucy loves the cinema and goes on average four times a week. The number of times she goes to the cinema in a week can be modelled as a Poisson distribution with a mean of four times.
Find the probability that Lucy goes to the cinema exactly five times in a week.
Find the probability that Lucy goes to the cinema no more than four times in a fortnight.
Find the probability that Lucy goes to the cinema at least once in a day.
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Comic Stans is a comic book store in the city of Krakoa. Customers enter the store randomly and independently at an average rate of 8 people every 15 minutes.
Find the probability that exactly three people enter the store in a 1-minute period.
Find the probability that someone enters the store in a 15-second period.
Find the probability that at most three people enter the store in a 10-minute period.
Find the variance of the number of people entering the store in a 1-hour period.
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Amber suggests that she can model the number of times that she hiccups using a Poisson distribution.
Write down two conditions that must apply for this model to be applicable.
The mean number of hiccups in a 30-second period is 2.9.
Assuming a Poisson distribution is applicable, find the probability that
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The table below shows the data from a sample of 50 observations of a variable .
Frequency |
The table below shows the data from a sample of 100 observations of a variable .
Frequency |
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Jim is a bird watcher and is trying to model the number of birds that fly past his window. During a 10-minute period he records the number of birds that fly past his window, and he repeats this a total of 120 times to form a sample.
Number of birds |
Frequency |
0 |
43 |
1 |
44 |
2 |
22 |
3 |
8 |
4 |
3 |
5 or more |
0 |
Calculate estimates for the mean and the variance for the number of birds that fly past Jim’s window in a ten-minute period.
Explain why a Poisson distribution would be appropriate to model the number of birds that fly past Jim’s window in a 10-minute period.
Jim uses the distribution to model the number of birds that fly past his window in a 10-minute period.
Use Jim’s model to calculate the probability that:
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The random variable .
Find:
Find the largest integer such .
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The random variable follows a Poisson distribution which has a standard deviation of 2.25.
Write down the value for
Find .
Find .
Find .
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Blue, a dog, barks an average of 25 times every hour. His owner, Hayley, uses a Poisson distribution to model the number of times that Blue barks.
Write down two assumptions Hayley has made about Blue’s barks in order to use a Poisson distribution.
Find the probability that Blue barks:
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Grace, a grumpy toddler, attends nursery five days a week. The number of tantrums that Grace has in a day follows a Poisson distribution with variance 3.14.
Find the probability that Grace has exactly 17 tantrums during a week at nursery.
Find the probability that Grace has fewer than four tantrums in a two-day period at nursery.
Given that Grace has fewer than four tantrums at nursery one day, find the probability that she had no tantrums at nursery that day.
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The table below shows the number of detentions per school day that Ms Ottway, a teacher, issues to students over a period of 150 days.
Number of detentions |
Number of days |
0 |
51 |
1 |
54 |
2 |
36 |
3 |
6 |
4 |
3 |
5 or more |
0 |
By calculating estimates for the mean and the variance, show that a Poisson distribution is an appropriate model for the number of detentions Ms Ottway issues.
Using a Poisson distribution with the estimated mean, find the probability that Ms Ottway issues at least 5 detentions in a day.
Students at Ms Ottway’s school attend school 5 days a week for 40 weeks a year.
Estimate the number of weeks in a school year that Ms Ottway issues fewer than 3 detentions.
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and are random variables with and .
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In a game, enemies appear independently and randomly at an average rate of 2.5 enemies every minute.
Find the probability that exactly 10 enemies will appear in a five-minute period.
Find the probability that at least 3 enemies will appear in a 90-second period.
The probability that at least one enemy appears in minutes is 0.999. Find the value of .
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The random variable . Find
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Matt receives text messages from his friends at an average rate of 4 messages per half hour.
State the assumptions needed to use a Poisson distribution to model the number of text messages Matt receives in a fixed time period.
Find the probability that
Matt’s friend, Jessica, bets that Matt will not receive more than text messages in the next hour. Jessica wants the probability of her losing the bet to be less than 5%.
Find the smallest value of that Jessica should use.
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Phoebe has a faulty smoke detector which rings at an average rate of 5 times a day.
Write down two conditions for a Poisson distribution to be a suitable model for the number of times that Phoebe’s smoke detector rings.
Given that Phoebe’s smoke detector rang at least once in a 6-hour period, find the probability that Phoebe’s smoke detector rang no more than 4 times during that period.
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The random variable
The random variable where is a positive integer.
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The random variables and .
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Roger is investigating historical data from World War II. In south London, an area of 144 km² was divided into 576 equal squares and it was recorded how many times each square was hit by a flying bomb over a period of time. During this time, it was recorded that a total of bombs landed in the area. The results are as shown in the table below:
Number of times hit by a flying bomb |
Number of squares |
0 |
|
1 |
|
2 |
|
3 |
|
4 |
|
5 or more |
Roger suggests that a Poisson distribution with mean of would be an appropriate model for the number of times a square was hit by a flying bomb.
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The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled as a Poisson distribution with mean of 1.2. The number of mistakes made by a different student, Qays, in a 30-minute revision period is modelled as a Poisson distribution with a mean of 2.2.
Find the probability that Priya makes exactly 2 mistakes and Qays makes exactly 1 mistake within a one-hour revision period. Write your answer in the form where and are integers to be found. State any assumptions that are needed.
Priya and Qays add together the number of mistakes they make after each revision session. Find the probability that they make a total of exactly 3 mistakes within a one-hour revision period. Write your answer in the form where and are integers to be found.
Given that Priya makes exactly 2 mistakes in a one-hour revision period, find the probability that Priya and Qays made exactly 3 mistakes in total in that revision period.
Given that Priya and Qays make exactly 3 mistakes in total in a one-hour revision period, find the probability that Priya made exactly 2 mistakes in that revision period.
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Whilst writing an essay, Gamu notices that she makes spelling mistakes at a rate of 7 for every 150 words. Gamu models the number of spelling mistakes she makes using a Poisson distribution.
Find the maximum number of words Gamu can write before the probability of her making a spelling mistake exceeds 0.75.
Gamu is asked to write three short essays by her lecturer. She writes one containing 100 words, one containing 200 words and one containing 250 words. An essay is returned by Gamu’s lecturer if more than 1% of its words contain spelling mistakes.
Find the probability that all three short essays are returned.
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Use the fact that is a valid probability distribution, along with the properties of the Poisson distribution, to demonstrate the validity of the Taylor series expansion
for all positive values of .
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Jenna manages a team of extraordinary scientists. The scientists work six days a week and make new discoveries at an average rate of one per week.
State any assumptions needed in order to use a Poisson distribution to model the number of new discoveries made by the scientists in a fixed time period.
Find the probability that
Jenna is planning a new project on which the scientists will work for weeks, where is an integer. Jenna wants there to be a probability of at least 1% that the number of new discoveries made by the scientists is more than double the number of weeks of the project.
Find the largest value of that Jenna should choose.
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