work1 2012

1. 1.1. In a card game, there are 52 cards, i.e. 13 different values, each of 4 different colours. You will obtain (randomly) 5 cards. Find the probability that a) you will obtain 4 kings (+ one other card). b) you will have all 5 cards of the same colour. 1.2. Lifetime of a battery is described by a random variable with the exponential distribution with EX=200 hours. Compute: a) What is the median of this distribution? b) probability that battery survives more than 100 hours. c) such time that 25\% of best batteries survive it (i.e. the 75% quantile of the lifetime distribution). 1.3. Analyze the following data (times to repair of air-conditions in Boeing planes, in hours of work): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution. 90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29, 118, 25, 156. From observed relative frequencies, estimate the probability that the next time to repair will be longer than 100 hours.

2. 2.1. Machine parts are tested. It is known that 1% of parts do not hold the test and break down. When we intend to test 200 such parts, what is the probability that more than 2 from them will break during the testing? (Use binomial distribution or Poisson approximation).

2.2. Let the weights of plastic parts vary according to Normal distribution N(mu=50g, sigma=0.7). Write the density function of this distribution. '1A' quality means that the weight is between 49 and 51 g. What is the proportion of 1A quality parts among all parts? 2.3. Analyze the following data (times to repair of air-conditions in Boeing planes, in hours of work): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution. 413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118, 34, 31, 18, 67, 57, 62, 7, 22, 34. From observed relative frequency, estimate the probability that the next time to repair will be shorter than 50 hours.

3. 3.1. 90 good products and 10 bad were mixed together. a) Estimate probability of bad product (i.e. prob. that one randomly selected product is bad). b). What is the probability that among 5 randomly selected products there are at least 4 good ones? 3.2. The heights of six-year old girls are normally distributed with the mean 117,80 cm and standard deviation 5,52 cm. Find the probability that a randomly selected six-years-old girl has height between 115 and 120 cm. 3.3. Analyze the following grouped data - table of incomes (ascertained by survey): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Plot the column graph (histogram) of the data. INCOME (class intervals), in 1000 CzK from 24 to 26 26 28 28 30 30 32 32 34 34 36

Number of people (observed frequency) 2 5 9 6 4 1

From observed relative frequencies, estimate the probability (proportion) of incomes larger than 30000 CzK. ---------------------------------------------------

4. 4.1. Imagine that you roll a die 30 times. Compute a) probability that there are exactly 5 results "6" b) probability that number of results "6" is greater than 5. 4.2. Lifetime (in hours) of a battery is described by a r. variable with Weibull distribution with parameters a = 300, b = 1.5, in notation with distribution function F(x)=1-exp(-(x/a)b), so that the mean lifetimel is about 271 hours. Compute 1. probability that a battery survives 300 hours; 2. such time T that 25% of batteries survive T (i. e. T is the 75% quantile of the distribution); 3. the point of maximal density (the modus). Write the density function and distribution function of this Weibull distribution 4.3. Analyze the following discrete-type data (observed numbers of accidents in 51 days): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Number of accidents frequencies 4 5 5 9 6 17 7 13 8 4 9 3 ------------------------------------From observed relative frequencies, estimate the probability that the next day no accident will occur.

5. 5.1. When it is assumed that male and female children births are equally probable, compute a) the probability of getting exactly six girls in ten births. b) the probability of more that four girls in seven births. 5.2. The weight of one product is a random variable with normal distribution N(µ = 200g; σ2 = 16). Find probability that 10 products together have weight less than 2020 g. 5.3. Analyze the following grouped data (upper blood pressures of a set of patients): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Plot the column graph (histogram) of the data. Blood presure intervals

Number of patients (observed frequencies)

from 100 to 105 6 105 110 13 110 115 22 115 120 19 120 125 11 125 130 5 130 135 2 --------------------------------------------------From observed relative frequencies, estimate the probability (proportion) of people with the blood pressure smaller than 110.

6. 6.1. 100 certain products were tested, 7 from them were bad, a) Estimate probability of bad product (i.e. prob. that one randomly selected product is bad). b). What is the probability that among 10 other randomly selected products there are at least 9 good ones? 6.2. Lifetime (in hours) of a battery is described by a r. variable with Normal distribution with parameters µ = 250 hours, variance σ2 = 900. What is the mean lifetime? Further compute, with the help of Excel (or from table of N(0,1) distribution function) 1. probability that battery survives 300 hours; 2. such time that 25% of batteries survive it (i. e. compute the 75% quantile of the distribution); Write the density function of this normal distribution.

6.3. Analyze the following data (times to failures of melting cells, in days of work): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution (i.e., is unimodal or uniform, symmetric or rather not,..). 468, 725, 838, 853, 965, 1139, 1142, 1304, 1317, 1427, 1554, 1658, 1764, 1776, 1990, 2010, 2224, 2280, 2371, 2541. Estimate the probability (proportion) of cells with lifetime between 1000 and 1500 days.

7. 7.1. In a 'miniloto', four numbers are drawn from numbers 1,. .,24. In your ticket, you also have to select four numbers. If you hit all 4 drawn numbers, your gain is 50 000 CzK, if you hit 3 numbers, you will obtain 1 000 CzK, even if you hit only 2 numbers, you will obtain 50 CzK. What is your expected (mean) gain from one ticket? [Compute first the probabilities of all 3 winning cases, then the mean gain = 50000*P(all 4 hit)+1000*P(3 hit)+50*P(2 hit). ] 7.2. Numbers of car accidents in last 8 weeks were 4,6,3,1,8,7,5,6. a) Estimate the mean and variance from these data. b) Provided data follows Poisson distribution of probability, What is the estimate of Poisson parameter lambda? c) From this Poisson distribution, compute then the probability that next week the number of accidents will be less than 3. 7.3. Analyze the following data (breaking strengths (in Kp) of nylon fibres): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution. 7.2240, 7.1030, 7.0750, 7.2400, 7.2320, 7.1720, 7.1070, 7.0910, 7.0790, 7.2520, 7.0710, 7.2320, 7.1190, 7.2520, 7.1560, 7.0230, 7.5370, 7.1030, 7.0030, 7.1190. Estimate the probability that the breaking strength of next fibre will be larger than 7.2 Kp.

8. 8.1. A study showed that among injured airline passengers, 47% of the injuries were caused by failure of the seat. a) Find the expectation EX, variance var(X) and standard deviation for the number of injured by seat failure from 200 injured passengers (consider binomial distribution). b) Compute probability that more than 100 injuries from 200 were caused by seat failures (here you can use also Normal approximation via the C.L.Theorem). 8.2. Lifetime of a battery is described by a random variable with the Exponential distribution with EX=100. Compute: a) What is the median of this distribution? b) probability that battery survives more than 150 hours. c) such time that 25% of best batteries survive it (i.e. the 75% quantile of the lifetime distribution). 8.3. Analyze the following data (breaking strengths (in Kp) of Cu wires): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution. 2.2550 2.2590 2.2230 2.2750 2.2230 2.2670 2.2950 2.2710 2.2390 2.2750 2.2710 2.2630 2.3070 2.2870 2.2670 2.2870 2.2470 2.2870 2.2630 2.2550 . Estimate the probability that the breaking strength of next wire will be smaller than 2.25 Kp.

9. 9.1. An insurance company (offering life insurance) insured 1000 men. Each man paid 120 Eur per year and in a case of death thein family will obtain 8000 Eur. If a probability of death of each man in one year is 0.008, compute the probability that during the zdar the company will be in loss (paying more than was obtained). 9.2. Lifetime (in hours of work) of an electronic device is described by a r. variable with Weibull distribution with parameters a = 200, b = 2, in notation with distribution function F(x)=1-exp(-(x/a)b), so that mean time to failure is about 177 hours. Compute 1. probability that device survives 200 hours; 2. such time T that 90% of batteries survive T (i. e. T is the 10% quantile of the distribution); 3. the point of maximal density (the modus). Write also the density function of this Weibull distribution. 9.3. Analyze the following data (two-months consumptions of el. energy in 31 residences, in KWh): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution. 728 774 859 882 791 731 838 862 880 831 759 774 832 816 860 856 787 715 752 778 829 792 908 714 839 752 834 818 835 751 837. Estimate the probability (proportion) of residences with consumption larger than 800 kWh in 2 months.

10. 10.1. Imagine that you roll a die 20 times. Compute: a) probability that exactly 4 results are "6". b) probability that the number of results "6" is smaller than 5. 10.2. Lifetime (in hours) of a battery is described by a r. variable with Normal distribution with parameters µ = 200 hours, variance σ2 = 1600. What is the mean lifetime? Further compute, with the help of Excel (or of table of N(0,1) distribution function) 1. probability that battery survives 250 hours; 2. such time that 25% of batteries survive it (i. e. compute the 75% quantile of the distribution); Write the density function of this normal distribution. 10.3. Analyze the following data (reaction times - in seconds - of 12 drivers to sudden appearence on an animal on the road): Compute the mean, variance, st. deviation, median, 25% and 75% quantile. Then plot the histogram and discuss the shape of distribution. 0.74 0.71 0.41 0.82 0.74 0.85 0.99 0.71 0.57 0.85 0.57 0.55 . Estimate the probability that the reaction time is shorter than 0.5 sec.

11. 11.1. In a card game, there are 52 cards, i.e. 13 different values, each of 4 different colours. You will obtain (randomly) 7 cards. Find the probability that a) you will obtain 4 kings (+ 3 other cards). b) you will have all 7 cards of the same colour. 11.2. The weight of one product is a random variable with normal distribution N(µ = 100g; σ2 = 9). Find probability that 20 products together have weight greater than 2020 g. 11.3. Analyze the following grouped data (frequencies of lottery numbers selected in a 26-week period): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Plot the column graph (histogram) of the data. Does they look as to be distributed uniformly? Numbers from 1 to 5 6 - 10 11 - 15 16 - 20 21 - 25 26 - 30 31 - 35 36 - 40

frequencies 19 24 18 21 23 20 16 15

From observed relative frequencies, estimate the probability that the next drawn numer will be greater than 30. ---------------------------------------------------

12. 12.1. Probability of occurrence of a success in one trial is 0,3. What is the probability that from 100 trials the number of successes will be between 20 and 40? (use binomial distribution or approximation with Central Limit Theorem). 12.2. Lifetime of a battery is described by a random variable with the exponential distribution with EX=150 hours. Compute: a) What is the median of this distribution? b) probability that battery survives more than 100 hours. c) such time that 25\% of best batteries survive it (i.e. the 75% quantile of the lifetime distribution). 12.3. When 40 people were surveyed at the shoping centre, they reported the distances they drove to the centre, and the values (in km) are given here 2 15 25 25

8 4 40 8

1 10 31 1

5 6 24 1

9 5 20 16

5 5 20 23

14 1 3 18

10 8 9 25

31 12 15 21

20 10 15 12

Compute: mean, variance, st. deviation, median, 25% and 75% quartiles, Further, plot histogram of the data and discuss the shape of distribution. From these data, estimate the probability (proportion) of customers coming from distance longer than 20 km.

13. 13.1. Machine parts are tested. It is known that 2% of parts do not hold the test and break down. When we intend to test 300 such parts, what is the probability that maximally than 3 from them will break during the testing? (Use binomial distribution or Poisson approximation).

13.2. The heights of six-year old boys are normally distributed with the mean 116 cm and standard deviation 6 cm. Find the probability that a randomly selected six-years-old boy has height between 110 and 115 cm. 13.3. Analyze the following grouped data (blood alcohol concentration of punished drivers): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Plot the column graph (histogram) of the data. Blood alcohol (%) Number of drivers (observed frequencies) from 0.8 to 0.10 9 0.10 0.12 16 0.12 0.14 12 0.14 0.16 3 0.16 0.18 1 --------------------------------------------------From observed relative frequencies, estimate the probability (proportion) of drivers having more than 0.15% alcohol concentration from those with more than 0.1%.

14. 14.1. Machine parts are tested. It is known that 3% of parts do not hold the test and break down. When we intend to test 100 such parts, what is the probability that more than 2 from them will break during the testing? (Use binomial distribution or Poisson approximation).

14.2. Lifetime (in hours) of a device is described by a r. variable with Normal distribution with parameters µ = 100 hours, variance σ2 = 400. What is the mean lifetime? Further compute, with the help Excel (or of table of N(0,1) distribution function) 1. probability that device survives 120 hours; 2. such time that 40% of devices survive it (i. e. compute the 60% quantile of the distribution); Write the density function of this normal distribution. 14.3. Following data are heights of 36 randomly selected students: 174, 178, 183, 168, 163, 175, 178, 177, 169, 182, 188, 176, 177, 178, 184, 185, 170, 168, 157, 158, 174, 174, 173, 171, 168, 170, 172, 174, 176, 179, 179, 188, 186, 181, 180, 169. Compute: mean, variance, st. deviation, median, 25% and 75% quartiles, Further, plot histogram of the data and discuss the shape of distribution. From data, estimate the proportion of students higher than 180 cm.

15. 15.1: 200 certain products were tested, 10 from them were bad. a) Estimate the probability of bad product (i.e. prob. that one randomly selected product is bad). b). What is the probability that among 10 randomly selected products there are at least 5 good ones? 15.2. The weight of one product is a random variable with normal distribution N(µ =100g; σ2 = 4). Find probability that 50 products together have weight less than 5020 g. 15.3. Analyze the following data (percentage of people with health problems in several areas close to highway): Estimate the mean, variance, st. deviation, median, 25% and 75% quantiles. Plot the column graph (histogram) of the data. 74,9 75,4 79,8 76,3 68,9 67,1 76,8 82,6 71,1 74,8 75,6 63,3 66,7 57,2 64,3 . ---------------------------------------------------

16. 16.1. In a small lottery, there are 200 tickets, 20 from them winning. If you have 5 tickets, Chat is the probability that at least one from your tickets will win?

16.2. Lifetime of a battery is described by a random variable with the exponential distribution with EX=300 hours. Compute: a) What is the median of this distribution? b) probability that battery survives more than 300 hours. c) such time that 25\% of best batteries survive it (i.e. the 75% quantile of the lifetime distribution). 16.3. Analyze the following data (the age of people working in a supermarket): Estimate the mean, variance, st. deviation, median, 25% and 75% quantiles. Plot the column graph (histogram) of the data. 24, 25, 19, 33, 21, 22, 43, 25, 37, 35, 20, 53, 60, 59, 22, 24, 19, 57, 59, 49. ---------------------------------------------------

17. 17.1. In a small lottery, there are 100 tickets, 15 from them winning. If you have 5 tickets, Chat is the probability that at least 2 from your tickets will win?

17.2. Lifetime (in hours) of a device is described by a r. variable with Normal distribution with parameters µ = 200 hours, variance σ2 = 400. What is the mean lifetime? Further compute, with the help Excel (or of table of N(0,1) distribution function) 1. probability that device survives 150 hours; 2. such time that 90% of devices survive it (i. e. compute the 10% quantile of the distribution); Write the density function of this normal distribution. 17.3. Analyze the following grouped data - table of incomes (ascertained by survey): Estimate the mean, variance, st. deviation, median, 25% and 75% quantile. Plot the column graph (histogram) of the data. INCOME (class intervals), in 1000 CzK from 24 to 26 26 28 28 30 30 32 32 34 34 36 36 38

Number of people (observed frequency) 3 4 10 16 14 5 2

From observed relative frequencies, estimate the probability (proportion) of incomes smaler than 30 000 CzK. ---------------------------------------------------

18. 18.1. Machine parts are tested. It is known that 2% of parts do not hold the test and break down. When we intend to test 150 such parts, what is the probability that maximally than 3 from them will break during the testing? (Use binomial distribution or Poisson approximation).

18.2. The weights of six-year old boys are normally distributed with the mean 25 kg and standard deviation 3kg. Find the probability that a randomly selected six-years-old boy has the weight over 27kg. 18.3. Analyze the following data (distribution of month incomes of the staff at an istitution, in thousand crowns): Estimate the mean, variance, st. deviation, median, 25% and 75% quantiles. Plot the column graph (histogram) of the data. 60, 51, 70, 52, 60, 43, 49, 51, 43, 46, 47, 34, 37, 39,38, 44, 35, 42, 31, 48. ---------------------------------------------------