statistics exercises

7/17/2019 Statistics Exercises

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Question 1

Consider an industrial process whose hourly number of breakdowns followa Poisson distribution with λ = 1.65 and the cost of each breakdown in $following a Normal distribution with mean 70 and variance (45)2.Calculate:

(i) The probability the exactly 3 hours will occur during three hours.

(ii) The average and standard deviation of the number of failures in aworking day of 8 hours.

(iii) The 10th and 90th centiles of waiting between two failures

(iv) The mean and standard deviation of the overall cost in a working day.

Question 2

Consider that the hourly number of calls arriving in a call-centre followa Poisson distribution with λ = 4.25 and the duration of each call follow aNormal distribution with mean 180 and variance (45)2 seconds. Calculate:

(i) The probability that exactly 8 calls will arrive in the next three hours.

(ii) The average and standard deviation of the number of calls arrivingin a period of 4 hours.

(iii) The 25th and 75th centiles of waiting time between two calls

(iv) The distribution of the overall duration of telephone calls in a periodof 4 hours.

Question 3

Let Y n be a sequence of independent Poision variables, such that the pa-rameter of the nth variable is n+2

2n .

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(i) write the probability distribution function of the first three variables

of the sequence.

(ii) Calculate the following probabilities;

(a) P (Y 2 = 3),

(b) P (Y 3 > 1).

(iii) Study the convergence of Y n in distribution and in quadratic mean.

(iv) Study the convergence of Z n = Y 1+Y n

2 in distribution and quadraticmean.

(v) Study the convergence of X n = Y n+Y n+1

2 in distribution and quadratic

mean.

Question 4

Consider that the lifespan of a battery (in hours) follow an exponentialdistribution with parameter κ = 0.005 and that four batteries can be dis-posed whose lifespans are i.i.d., and using them each once a time to keep adesk lamp functioning.

(i) Write the density function of the overall duration X of the desk lamp.

(ii) Compute E (X ) and S (X ).

(iii) Suppose the first battery breaks down after 300 hours. How does theexpectation of (X ) change?

(iv) Calculate the probability that the last battery used will last longer

than the other ones.

(v) How long will the desk lamp be functioning if every battery has amedian duration?

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Question 5

Let X 1, X 2, andX 3 be three i.i.d., exponential variables, with parameterκ = .03.

(i) Denote with X [1], X [2], X [3] the corresponding order statistics.

(ii) Write the c.d.f of;

(a) X [1],

(b) X [2],

(c) X [3].

Question 7

Let X and Y be two independent discrete random variables such that:

P (X = −2) = 0.25; P (X = −1) = 0.40; P (X = +2) = 0.35.

P (Y = −1) = 0.25; P (X = +1) = 0.75.

Consider the random variables; T = X + Y + 1, andU = X − Y − 1.

(a) Represent the joint distribution of (T, U ) and marginal distributionin a two way table.

(b) Determine the independence table and check whether T and U areindependent or not.

(c) Calculate a chi-squared measure of association and normalize it.

(d) Calculate;

The expectations, E (T ) and E (U ) and the variances, V (T ) and V (U ).

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Question 8

Let X and Y be two independent discrete random variables such that:

P (X = −1) = 0.10; P (X = 0) = 0.60; P (X = +2) = 0.30.

P (Y = 0) = 0.35; P (X = +1) = 0.40; P (X = +3) = 0.25.

Consider the random variables; T = X + Y , and U = X − Y .

(a) Represent the joint distribution of (T, U ) and marginal distribution

in a two way table.

(b) Determine the independence table and check whether T and U areindependent or not.

(c) Calculate a chi-squared measure of association and normalize it.

(d) Write the condition distribution of U given (T = 3)

(e) Write the condition distribution of T given (U > 3)

(f) Calculate the expectations, E (T ) and E (U ),the variances, V (T ) andV (U ),the covariance Cov(T, U ).

Question 9

Consider the event that a die without the number (5) and instead with thenumber (4) in two of its faces i.e {1,2,3,4,4,6},is rolled. Let X 1, X 2, X 3, X 4be the number of points in four independent trials. Given the transformationT = X 1 + X 2 + X 3 + X 4, calculate:

(i) P (T = 13),

(ii)P (T > 11|X 1 = 4)

(iii) The expected value E (T ) and variance V (T ).

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(iv) What is the probability distribution of U = min{X 1, X 2, X 3, X 4} and

W = max{X 1, X 2, X 3, X 4}.

(v) Calculate M e(U ), E (U ), S (U ), E (W ), andV (W ).

Question 10

Consider the game in which a player rolls a die and tosses as many coinsas the number of die points, with a maximum of 4 coins. The first coin tossedis a 0.50c coin and the 2nd and 3rd are 1 euro coins, the 4th coin is valued 2euros. The player wins all the coins showing a tail. Let X be the number of

coins tossed and Y the final sum that the player wins.

(i) Calculate E (X ), V (X ), and M e(X ),

(ii)Calculate E (Y ), and V (Y ),

(iii) Calculate the following conditional probabilities;

(a) P (Y = 2|X = 3)

(b) P (X = 3|Y = 2)

(iv) Calculate the covariance and linear correlation coefficient of X and Y .

Question 11

Let R be the number of points observed after rolling a red die, and B thenumber of points observed after rolling a blue die. Let T = max{R, B} andU = (R − 1)(B − 1).

(i) Calculate the probability function, the expected value and variance of

T .

(ii) Calculate the probability function, the expected value and varianceof U .

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(iii) Calculate the following probabilities;

(a) P (U > 0)

(b) P (U > T )

(c) P (U > 5|T = 3)

(iv) Calculate the covariance and the linear correlation coefficient betweenT and U .

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