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Unformatted text preview: Math 521 Fall 2001 Final Exam Problem Points Score Name:
(Please Print.) Do all your work on this exam. Correct answers must be supported by your calculations and reasoning where appropriate. 1 . I have a drawer with 38 white socks and 62 green socks. If I put
on two socks at random, what is the probability that I am wearing a matching pair? 2 . Ten friends, including Nadine and Bernie, go to see a movie. They sit in 10 adjacent seats in one row. What is the probability that Nadine and Bernie sit beside each other? 3 . A store has 2 employees. Employee A serves 60% of the customers and enters an incorrect price for
1% of those customers. Employee B serves 40% and enters an incorrect price for 3% of those customers. If a customer complains to the manager because of an incorrect price charged, what is the
probability that employee B served the customer? Fun Question: Match the holiday celebration with the country. Luciadagen_ Duvali _ Kakizone __ Feast ofthc Radishes _ Ynlda (21) India (b) Mexico (c) Iran (d) Japan (e) chcdcn i “to s s 2
4 . Let X be a random variable with probability density function f(x) = {2 x l X 0 otherwise
(a) Compute the probability distribution ofX. (b) Compute the probability distribution function of Y = X2 + l. (c) Compute the probability density function of Y = X2 + l. 5. Let X and Y bejointly distributed random variables with joint probability density function 1 if0<y<2x, 0<x<l
0 otherwise Mei (a) Sketch the region where f(x,y) is nonzero. (b) Compute the marginal probability density function fX(x). (c) Compute the conditional probability density function fy,X(y l x). 6. Let X be the random variable with probability mass function f{x)=pq"_l, for x: l,2,3...and f(x) = 0 otherwise. Compute the moment generating function of X. (Hint: After a little algebra you should be able to use Ear)“l = l a if lrl<l.)
—r .\'=1 7 . A multiplex movie theatre has 6 movies, including Halloween X. playing. Patrons randomly and independently choose a movie to see. If 20 patrons attend on a panicular day, what is the probability
' that at least 2 of them have to suffer through Halloween X? 8 . The Bemoullis host the Newton family for Thanksgiving. Twelve Bemoullis and 8 Newtons are present. Unfortunately the Bernollis’ pet cat has contaminated three of the twenty dinner plates with
salmonella. The plates are distributed at random. (21) What is the probability that only Newtons get food poisoning? (b) What is the probability that at least two Newtons get sick? (3) State the Central Limit Theorem. (b) Assnme X is a binomial random variable with parameters n = 100 and p = 0.4. Use the central limit
theorem to approximate Pr[X S 44]. (Recall that we get a binomial random variable with parameters It andp by adding 22 independent Bernoulli random variables with parameter p. If needed, a table of values for the standard normal distribution is reprinted below.) Reprinted from Text: 570 Appendix of Tables 1 I TABLE A [Standard Normal Distribution Function: (13(1) = ﬁ If 2 —°o
x .00 .01 .02 .03 .04 .05 06 .07 .08 .09
.0 5000 5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
. l 5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
.2 5793 .5832 .5871 59/0 .5948 .5987 .6026 .6064 .6103 .614]
.3 .6179 .62/7 .6255 .6293 .6331 .6368 .6406 6443 6480 .6517
.4 .6554 6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
.5 6915 6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6 .7257 7291 .7324 .7357 .7389 .7422 .7454 .7486 7517 .7549
.7 7580 7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 7852
.8 788/ .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9 8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
[.1 .8643 8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
[.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 9761 9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 . 9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 . 9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2. 9 .998] .9982 . 9982 .9983 .9984 .9984 .9985 .9985 .9986 9986
3.0 .9987 . 9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 . 9990
3.] 9990 9991 9991 .9991 .9992 .9992 9992 .9992 9993 .9993
3.2 9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 9995
3.3 9995 .9995 9995 .9996 .9996 .9996 .9996 9996 9996 .9997
3.4 .9997 9997 9997 9997 9997 9997 9997 9997 9997 9998 ...
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