Download Probability and statistics 1 Random variables 2 Special discrete

Probability and statistics
practice sheet
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Random variables
1.1. Malév Hungarian Airlines operates two flights between Budapest and London daily. The probability that there
are no empty seats on the morning flight is 0.9; whereas 0.6 is the probability that there are no seats left on the
evening flight. Choosing a day randomly let ξ mean the number of those flights on which there are no empty
seats. Give the distribution of ξ and calculate its expected value and variance.
1.2. On an exam of an easy subject, examinees can put the chosen topic back at most twice, if they do not like it.
A student has learnt the 43 of the topics, and puts the topics back until he draws one, which he has learnt (or
he has no more opportunities to do so). Let ξ be number of necessary draws. Give the distribution of ξ, and
calculate its expected value and standard deviation.
1.3. There are six topics A and six topics B on an exam; one of each group has to be drawn. A student has not
learnt topics 3 and 6 in either group of topics. Let ξ be the number of those topics drawn, which he has not
learnt. Give the distribution of ξ, and calculate its expected value and standard deviation.
1.4. Two drawers out of four contain an object. We pull them out until we find any of them. Let ξ mean the number
of tries. Give the distribution of ξ, and calculate its expected value and standard deviation.
1.5. Let us suppose that a customer finds the sought article in a shop with probability 0.8. When he goes to get this
article and finds it in the first shop, he obviously stops shopping, but if not, he goes in another one where he
might find it. He decides to go in at most 4 shops. Let ξ mean the number of shops the customer went in. Give
the distribution of ξ, and calculate its expected value and standard deviation.
1.6. Two gamblers play the following game: player A rolls a die and pays twice as much Ft’s to player B as the
number on the die. Player B pays 6 Ft to A after each roll. For whom is the game profitable, i. e whose expected
gain is greater?
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Special discrete random variables
2.1. Fifteen percent of people are left-handed.
(a) Calculate the probability that the number of left-handed people among ten randomly chosen ones is twice
the expected.
(b) How many people should be chosen so that the probability that there is at least one left-handed among
them is at least 95%.
2.2. In Hungary 15% of people speak English. During a survey 25 people are asked if they speak English.
(a) Calculate the probability that 5 people say they speak English.
(b) Calculate the probability that the number of English-speakers deviates from the expected value by at most
one.
2.3. What is the highest allowed ratio of defectives, if we want the probability that there are no defectives in a 5-piece
sample to be at least 95%?
2.4. A cruel teacher has learnt that 60% of his class had not prepared for his lesson. How many pupils should he ask,
if he wants that the probability that there are at least one pupil among the chosen ones who had not prepared,
to be at least 90%?
2.5. Watching the cars passing in front of our house we counted 75.
(a) Calculate the probability that in a certain minute only two cars pass.
(b) Calculate the probability that in five minutes’ time at least three cars pass in front of our house.
2.6. Someone observed that every night the owl nesting on the tree in front of his window hoots once every 30 minutes
on average. What is the probability that
(a) during the next 30 minutes he hears more than one hoots;
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(b) during the next hour the owl hoots twice as many as the expected?
2.7. How many misprints are allowed in a 400-page book, if we want the probability that there are no misprints on
a certain page to be at least 95%?
2.8. A teacher observed that the probability that he fails at least one student on an exam is 0.8647.
(a) How many students does he fail during an exam on average?
(b) What is the probability that he fails twice as many students on an exam as he would expectedly do so
during two exams?
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Distribution and density functions of continuous random variables
3.1. Let F (x) =
1
1+e−x .
(a) Prove that F (x) can be the distribution function of a random variable ξ.
(b) Give the density function of ξ.
(c) Calculate the probablilty P (ξ ≥ ln 2).
√
3.2. Function F (x) = c x (0 < x < 4) is given, and it is constant for all other values of x.
(a) Give F , so that it is the distribution function of a continuous random variable ξ.
(b) Give the density function of ξ.
(c) Calculate the probability P (0 < ξ < 1), and indicate it on the graphs of F and f .
3.3. The following function is given:
f (x) =
(a)
(b)
(c)
(d)
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A 1 − x2 , if − 1 < x < 1;
0,
otherwise.
Determine the value of A, so that f (x) is the density function of a random variable ξ.
Give the distribution function of ξ.
Calculate the probability P − 21 < ξ < 12 .
Calculate the expected value and the variance of ξ.
Special continuous random variables
4.1. The nominal mass of the hazelnut chocolate bar Boci is 87.5 g. The actual mass can be considered a normally
distributed random variable with expected value 87.5 g, and standard deviation 0.8 g.
(a) What is the probability that the mass of a randomly chosen bar is greater than 88.5 g?
(b) During a check 300 bars were examined. It was found that the mass of 9 bars were less than 87 g, and the
mass of 273 bars were less than 89 g. What is the expected value and standard deviation?
4.2. The quantity of soft drinks filled in two-litre bottles can be considered such a normally distributed random
variable, whose expected value is 2 l, and whose standard deviation is 0.02 l.
(a) Calculate the probability that the quantity filled in a certain bottle is less than 1.97 l.
(b) How should the standard deviation of the filling machine be modified, so that the probability that the
quantity of the soft drink in a bottle is between 1.97 l and 2.03 l is at least 95%.
4.3. The quantity of beer filled in half-litre bottles can be considered such a normally distributed random variable,
whose expected value is 0.5 l, and whose standard deviation is 0.01 l.
(a) Calculate the probability that the quantity filled in a certain bottle is between 0.48 l and 0.52 l.
(b) How should the expected value of the filling machine be modified, so that the probability that the quantity
of the beer in a bottle is at least half a litre, is at least 95%.
4.4. The current price of a certain share can be considered such a normally distributed random variable, whose
expected value is 1500 Ft, and whose standard deviation is 30 Ft.
(a) How many percent of the shares were sold for less than 1450 Ft on this day?
(b) What is the probability that the deviation of the current price from the expected value is less than 1%?
(c) What deviation is expected from the expected value with 85% probability?
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Investigation of hypotheses
5.1. We asked 6 students how much coffee they drank while they were preparing for a comprehensive examination.
They gave the following answers (in litres): 1.3, 0.0, 0.8, 1.4, 0.9, 0.7. The quantity of the coffee consumed can
be considered a normally distributed random variable. Based on this sample, can we state that sudents drank
more than half a litre of coffee on average? Decide at 5% significance level.
5.2. Someone tests how many strikes they can make on the keyboard in a minute. They get the following data: 221,
198, 210, 190, 202, 208, 199. The number of strikes can be considered a nomally distributed random variable.
Based on this sample, can we say that the expected value of the number of strikes is at least 200? Decide at 5%
significance level.
During another test they count how many strikes they can make after drinking 0, 1, 2, 3, 4 pints of beer.
Measured data are, respectively: 205, 192, 153, 140, 110. Show that there is a strong linear correlation between
the quantity of the consumed beer and the number of strikes. Give the equation of the regression line.
5.3. The mass of a certain chocolate bar can be considered a normally distributed random variable with known
standard deviation. A sample of 10 was taken, and the 95% confidence interval had unit length.
(a) What is the standard deviation of the machine which packs the bars?
(b) Give an estimate for the expected value, if the confidence interval was (19.2, 20.2).
(c) Based on this sample, can we state that the machine packs significantly lighter chocolate bars than the
pre-adjusted 20 g? Decide at 5% significance level.
5.4. The fruit-content of fruit-juices can be considered a normally distributed random variable. It has to be established
whether a certain type of juice contains the declared 50% fruit or not. Twenty boxes are examined and the
20
20
P
P
following percentage data are received:
xi = 976 and
x2i = 47650. Decide at 5% significance level whether
i=1
i=1
the deviation from the official value is significant.
5.5. The weight of the flour filled in paper bags can be considered a normally distributed random variable. It has to
be established whether the filling machine still makes 2-kg packets or it has been misadjusted. Thirty bags are
30
30
P
P
examined and the following data are received:
xi = 58.2 and
x2i = 113.7. Decide at 5% significance level
i=1
i=1
whether the expected deviation from the official value is significant.
5.6. The running-time of the InterCity train berween Budapest and Szeged is 2.5 hours. The actual running-time
is measured 6 times, and the following data are received: 2.4, 2.7, 2.5, 2.6, 2.8, 2.6. The running-time can be
considered a normally distributed random variable. Based on this sample, can we say that the train is delayed?
Decide at 5% significance level.
5.7. The owner of a small grocery shop orders 50 litres of milk each day, but it seems that he could sell more.
Therefore he orders a larger amount for a week and records the sold quantity: 53, 47, 55, 52, 56, 55, 53. Is it
worth for him to order more milk, i. e. based on theses data, can he say that the amount of milk he can sell has
increased significantly?
5.8. A worker records how many hours he has worked 10 times, and he gets the following data (in hours): 8.5, 8.0,
8.1, 7.6, 8.7, 7.9, 7.8, 8.5, 9.0, 7.9. Working time can be considered a normally distributed random variable. Can
the worker state that he works more than 8 hours? Decide at 5% significance level.
5.9. The following table contains data concerning the concentration of pollution measured at 10 busy junctions on
two different days. Has air-pollution changed signifcantly? Decide at 5% significance level.
1st day
2nd day
20.9
21.4
17.1
16.7
15.8
16.4
18.8
19.2
20.1
19.9
15.6
16.6
14.8
15.0
24.1
24.0
18.9
19.2
12.5
13.2
5.10. In a paper-mill A4 papers are packed in 500-piece packets. The number of sheets in a packet can be considered
a normally distributed random variable, whose expected value is 500.
(a) What is the standard deviation, if the probability that the number of sheets in a packet is less than 490, is
2%?
(b) What is the probability that choosing ten packets randomly, there is one which contains less than 490
sheets?
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(c) During an investigation they want to check if the packets contain 500 sheets or significantly less. To do
this, the contents of five packets are counted, and the following numbers are received: 490, 495, 502, 499,
494. Decide at 5% significance level.
5.11. The following piece of information was announced on Radio Juventus on 4th January 2002: “Polish scientists
established that the Belgian 1 euro coin is biased, or ‘head-favoured’. The coin was tossed 250 times, and landed
with King Albert’s face on top 140 times.” Is the difference really significant? Decide at 5% significance level.
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