MA2631-Assignment 10 Solved

1. Assume that X is a normally distributed random variable with mean µ and variance σ2. Compute the probabilities that X is not more than one, two and three standard deviations away from the mean, i.e.  and

 . The other way round, how you have to choose k such that X stays
with 95% in the interval (µ − k,µ + k)? How about 99%?

2.    Let X be a standard normal distributed random variable. How we have to choose β ∈ R such that         5?

3.    Let X ∼ N(0,1) be a standard normal distributed random variable. Calculate the moment generating function mX(λ) = E[eλX] for λ ∈ R. Use the moment generating function to calculate mean and variance of X, confirming what we know already.

4.    An Airline sold 560 tickets for an Airbus 380 flight (capacity: 555 seats) in the assumption that not all passengers that bought a ticket will arrive for the flight. Assume that the probability that a passenger will not shop up for the flight is 1%, independently for all passengers. How likely is it that there are more passengers showing up for the flight than seats are available? Calculate this probability by using

a) a binomial distribution for the number of passengers that showed up for the flight;
b) a normal approximation.

5.    Let X by a standard normal distributed random variable. Calculate E[Xn] for an arbitrary non-negative integer n.

Hint: Consider the cases of odd and even n separately and try to express (via integration by part) E[Xn+2] in terms of E[Xn]. You should then be able to make an educate guess it prove it by induction.

6.    A random variable X is called log-normal with parameters µ and σ, if X = eY where Y ∼ N(µ,σ2).

a)     Express the cdf FX and the density fX of X in terms of density ϕ and cdf Φ of a standard normal variable.

b)    What are expectation and variance of X?

c)     Let now µ = 0 and σ = 1. Calculate P[X > 2] and find α such that P[X ≤ α] = 99%.

8 points per problems
Additional practice problems (purely voluntary - no points, no credit, no grading):

Standard Carlton and Devore, Section 3.3: Exercises 40, 43, 44, 47, 70;

Challenging Assume that the time between the arrivals of processes at a server is exponentially distributed with parameter λ and the waiting times are independent. Denote by Nt the number of processes which arrived up to time t. Calculate (for fixed t > 0) the distribution of the random variable Nt.

Table of the cdf of the standard normal distribution Φ(x)

                                               x            0.00         0.01         0.02         0.03         0.04         0.05         0.06         0.07         0.08         0.09

                                            0.0        0.5000     0.5040     0.5080     0.5120     0.5160     0.5199     0.5239     0.5279     0.5319     0.5359

                                            0.1        0.5398     0.5438     0.5478     0.5517     0.5557     0.5596     0.5636     0.5675     0.5714     0.5753

                                             0.2        0.5793     0.5832     0.5871     0.5910     0.5948     0.5987     0.6026     0.6064     0.6103     0.6141

                                            0.3        0.6179     0.6217     0.6255     0.6293     0.6331     0.6368     0.6406     0.6443     0.6480     0.6517

                                            0.4        0.6554     0.6591     0.6628     0.6664     0.6700     0.6736     0.6772     0.6808     0.6844     0.6879

                                            0.5        0.6915     0.6950     0.6985     0.7019     0.7054     0.7088     0.7123     0.7157     0.7190     0.7224

                                            0.6        0.7257     0.7291     0.7324     0.7357     0.7389     0.7422     0.7454     0.7486     0.7517     0.7549

                                            0.7        0.7580     0.7611     0.7642     0.7673     0.7703     0.7734     0.7764     0.7794     0.7823     0.7852

                                            0.8        0.7881     0.7910     0.7939     0.7967     0.7995     0.8023     0.8051     0.8078     0.8106     0.8133

                                            0.9        0.8159     0.8186     0.8212     0.8238     0.8264     0.8289     0.8315     0.8340     0.8365     0.8389

                                            1.0        0.8413     0.8438     0.8461     0.8485     0.8508     0.8531     0.8554     0.8577     0.8599     0.8621

                                            1.1        0.8643     0.8665     0.8686     0.8708     0.8729     0.8749     0.8770     0.8790     0.8810     0.8830

                                            1.2        0.8849     0.8869     0.8888     0.8907     0.8925     0.8944     0.8962     0.8980     0.8997     0.9015

                                            1.3        0.9032     0.9049     0.9066     0.9082     0.9099     0.9115     0.9131     0.9147     0.9162     0.9177

                                            1.4        0.9192     0.9207     0.9222     0.9236     0.9251     0.9265     0.9279     0.9292     0.9306     0.9319

                                            1.5        0.9332     0.9345     0.9357     0.9370     0.9382     0.9394     0.9406     0.9418     0.9429     0.9441

                                            1.6        0.9452     0.9463     0.9474     0.9484     0.9495     0.9505     0.9515     0.9525     0.9535     0.9545

                                            1.7        0.9554     0.9564     0.9573     0.9582     0.9591     0.9599     0.9608     0.9616     0.9625     0.9633

                                            1.8        0.9641     0.9649     0.9656     0.9664     0.9671     0.9678     0.9686     0.9693     0.9699     0.9706

                                            1.9        0.9713     0.9719     0.9726     0.9732     0.9738     0.9744     0.9750     0.9756     0.9761     0.9767

                                            2.0        0.9772     0.9778     0.9783     0.9788     0.9793     0.9798     0.9803     0.9808     0.9812     0.9817

                                            2.1        0.9821     0.9826     0.9830     0.9834     0.9838     0.9842     0.9846     0.9850     0.9854     0.9857

                                            2.2        0.9861     0.9864     0.9868     0.9871     0.9875     0.9878     0.9881     0.9884     0.9887     0.9890

                                            2.3        0.9893     0.9896     0.9898     0.9901     0.9904     0.9906     0.9909     0.9911     0.9913     0.9916

                                            2.4        0.9918     0.9920     0.9922     0.9925     0.9927     0.9929     0.9931     0.9932     0.9934     0.9936

                                            2.5        0.9938     0.9940     0.9941     0.9943     0.9945     0.9946     0.9948     0.9949     0.9951     0.9952

                                            2.6        0.9953     0.9955     0.9956     0.9957     0.9959     0.9960     0.9961     0.9962     0.9963     0.9964

                                            2.7        0.9965     0.9966     0.9967     0.9968     0.9969     0.9970     0.9971     0.9972     0.9973     0.9974

                                            2.8        0.9974     0.9975     0.9976     0.9977     0.9977     0.9978     0.9979     0.9979     0.9980     0.9981

                                            2.9        0.9981     0.9982     0.9982     0.9983     0.9984     0.9984     0.9985     0.9985     0.9986     0.9986

                                            3.0        0.9987     0.9987     0.9987     0.9988     0.9988     0.9989     0.9989     0.9989     0.9990     0.9990

                                            3.1        0.9990     0.9991     0.9991     0.9991     0.9992     0.9992     0.9992     0.9992     0.9993     0.9993

                                            3.2        0.9993     0.9993     0.9994     0.9994     0.9994     0.9994     0.9994     0.9995     0.9995     0.9995

                                            3.3        0.9995     0.9995     0.9995     0.9996     0.9996     0.9996     0.9996     0.9996     0.9996     0.9997

                                            3.4        0.9997     0.9997     0.9997     0.9997     0.9997     0.9997     0.9997     0.9997     0.9997     0.9998

                                            3.5        0.9998     0.9998     0.9998     0.9998     0.9998     0.9998     0.9998     0.9998     0.9998     0.9998

                                            3.6        0.9998     0.9998     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999

                                            3.7        0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999

                                            3.8        0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999     0.9999

                                            3.9        1.0000     1.0000     1.0000     1.0000     1.0000     1.0000     1.0000     1.0000     1.0000     1.0000

Φ(x) x

For negative values of x we just use the identity Φ(x) = 1 − Φ(−x). Table 5.1: Table for the cumulative distribution function of a standard normal distribution

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