## expected value problems and solutions pdf

0000002372 00000 n Fully worked-out solutions of these problems are also given, but of course you should ﬁrst try to solve the problems on your own! 0000080456 00000 n 0000001558 00000 n %%EOF 1420 0 obj<>stream 0000000873 00000 n <]>> What does the P(X = x) column sum to? 1417 0 obj <> endobj 0000090189 00000 n Total experiments (N) = 30. In another class of problems, it is possible to acquireŒoften at a certain Solution. 0000003621 00000 n Expected Value Deﬁnition 6.1 Let X be a numerically-valued discrete random variable with sam-ple space Ω and distribution function m(x). 0000022636 00000 n a. 205.) For each of the following probability distributions, calculate the expected value and standard deviation: a. 89 23 0000003272 00000 n 0000001243 00000 n 0000079279 00000 n E = expected value of winnings - cost of game Outcomes Probability Winning of sCtso game Win 0.659 $3 $2 Lose 0.341 $0. 0000002012 00000 n Therefore, the expected value of getting blue ball is approximately 8. 0000001213 00000 n 0000022357 00000 n • What is the expected value of your investment? The table should have two columns labeled x and P(X = x). Compute the expected length of the arc that contains the point (1,0). side. Gamblers wanted to know their expected long-run winnings (or losings) if they played a game repeatedly. 111 0 obj <>stream 0000000016 00000 n ECE302 Spring 2015 HW9 Solutions May 3, 2015 2 (b)Find the conditional expected value, E[YjA]. 0000007457 00000 n 0000004484 00000 n 0000003747 00000 n If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. the formula of expected value, and compute the expected value of a game, Statistics and Probability, how to find expected value of an event, Conditional probability and expected values, examples and step by step solutions, HSS-MD.A.3, random variable, expected value <<9677f80f1c0983448a384dbcca936b1d>]>> 0 %PDF-1.6 %���� . 0000079882 00000 n Risk measurement 1. Risk measurement 1. 0000009547 00000 n Here is a “solution.” Interpret them as cuts that divide the circle into three arcs. startxref X takes on what values? Let y be the amount of his fee. ��ٌ��w+O�4�V���ϕ�4���z�^�1����ɫ��!����L2�3m���X 2-�1��S�MF1��w�Ϊ��f�E�N�� bi~B�( 2��5�ѩ@Z�u�"J���F�_�o�g0h���Rþ^�!BaM�P�%���n��E��;71�X��6A`AP�\��{�K8"/]S�gb�g�``����. 0000003870 00000 n The expected value of the game is given by 12 2 12 20 8 30 12 3 10* 12 4 2* 12 5 ( ) 4* = − + − = E X = + − You should expect to lose $16.67 after one-hundred games. 0000001436 00000 n 0 startxref E(X) = µ. 0000003878 00000 n 0000022036 00000 n This term has been retained in Find out the expected value of its outcomes. Find the sum of the elements of S. The next three problems are harder; in these problems linearity of expectation is not the main idea of the solution. 0000051634 00000 n Solutions to risk and return practice problems 1 . 0000090582 00000 n Solutions to risk and return practice problems 1 . We often refer to the expected value as the mean, and denote E(X) by µ for short. +p(On)Xn The expected value is often denoted just by E. 8 Calculating Expected Value E = + + = 11 1 273 15()-42 4 4 − Make a table like this one Outcome Probability Value of X HH 1/4 7 HT or TH 1/2 3 TT 1/4 -15 O1 O2 O3 X1 X2 X3. trailer Let Sbe the set of real numbers that can be represented as repeating decimals of the form 0:abcwhere a;b;care distinct digits. calculating the expected monetary values of all alternatives, and selecting the alternative with the highest expected monetary value. the highest expected proÞt or revenue, or the lowest expected cost. x���1 0ð4\ 0000007867 00000 n As Hays notes, the idea of the expectation of a random variable began with probability theory in games of chance. 0000001420 00000 n 0000000016 00000 n Answer to Complete the PDF and answer the questions. Problem 3 (Solution on p. xref Probability Exam Questions with Solutions by Henk Tijms1 December 15, 2013 This note gives a large number of exam problems for a ﬁrst course in prob-ability. 0000008875 00000 n 0000052373 00000 n 0000038699 00000 n Expected value = Number of experiments × Probability = N × p = 30 × 0.25 = 7.50. Find the expected value of S. Problem 2.5 (AIME 2006 #6). x�b```"cV6'Ad`B��������aM�~�\����Rxc�K��1n�ʌ+lF�g�700��ŦI$�^�eȠ�S�茉s�CH�OjH$�[p��"�-ctی4����S"�*Lj�l��X�s����6� 0000003371 00000 n 0000001531 00000 n Outcome Probability 2Outcome value px x-E(x) (x-E(x))2 p(x-E(x)) Good 30% $40 $12$16$256 77 Normal 50% $20 $10-$4$16 8 Bad 20% $10 $2-$14$196 39 100% E(x) = $24 variance = 124 standard deviation = $11 b. Problem 7.2.4 Solution From De nition 3.6, the PDF of Y is xref For a major accident, the policy pays $5000; for a minor accident, Find the expected value. X and Y are dependent), the conditional expectation of X given the value of Y will be diﬀerent from the overall expectation of X. f Y (y)= y/20≤ y ≤ 2 0 otherwise (1) The expectation is E[Y]= 2 0 y2 2 dy =4/3(2) To ﬁnd the variance, we ﬁrst ﬁnd the second moment E Y2 = 2 0 y3 2 dy =2. 4 Chapter 3: Decision theory be interpreted as the long-run relative frequencies, and theexpected payo ﬀ as the average payo ﬀ in the long run. Example 1: Researchers have conducted a survey of 1600 co ee drinkers asking how much co ee they drink in order to con rm previous studies. Expected value Investment problem: • You have 100 dollars and can invest into a stock. Probability, Expectation Value and Uncertainty We have seen that the physically observable properties of a quantum system are represented by Hermitean operators (also referred to as ‘observables’) such that the eigenvalues of the operator represents all the possible results that could be obtained if the associated physical observable were to be measured. trailer 4. 0000003948 00000 n 0000012948 00000 n 0000006050 00000 n Construct a probability distribution table (called a PDFtable) like the one in the previous example. %%EOF 0000053096 00000 n These problems will tend to be of the repetitive kind, so that the probabilities may. 0000000756 00000 n ���s��6?Y��V#P9l`�'��RK�ӽ�N��_�RV*�yU���[���\��C�X��9�Ɂ�KzN�ْ�P�+��YJ����� � �V"�P�uw��0l0 0000003333 00000 n 18. Compute the expected value given a set of outcomes, probabilities, and payoffs If you're seeing this message, it means we're having trouble loading external resources on our website. 0000001306 00000 n 89 0 obj <> endobj 0000005336 00000 n (3) The variance is then Var[Y]=E[Y2]−E[Y]2 =2−(4/3)2 =2/9. In this problem we want to determine the detective’s fee so that the expected value is zero. The determina-tion of the optimal alternative becomes a little more complicated when the alternatives involve sequences of decisions. (vii)Reject H 0 if ˜2 is larger than the critical value (right-tailed test). Example 6.17. The returns are volatile and you may get either $120 with probability of 0.4, or $90 with probability 0.6. A fair die is thrown. 0000008829 00000 n 0000001869 00000 n The expected value of a random variable is the arithmetic mean of that variable, i.e. 0000010277 00000 n Show that the expected number of components is greater than mn/8 and smaller than (m+2)(n +2)/6. If the value of Y aﬀects the value of X (i.e. 0000006738 00000 n The expected value of the random variable X is, by definition: E(X) = p(O1)X1 + p(O2)X2 +p(O3)X3 + …. Find the probability that x = 2. b. %PDF-1.5 %���� Remark. 3. endstream endobj 1443 0 obj<>/W[1 1 1]/Type/XRef/Index[56 1361]>>stream First-step analysis for calculating the expected amount of time needed to reach a particular state in a process (e.g. {\G����M�ѕ�y�C. Probability of getting a blue ball = (p) = 1/4 = 0.25. c 2013 by Henk Tijms, Vrije University, Amsterdam. A. the expected … 4 13 Insurance Example • An insurance company charges $150 for a policy that will pay for at most one accident. Problem 3.3.4 Solution We can ﬁnd the expected value of X by direct integration of the given PDF. Outcome Probability 2Outcome value px x-E(x) (x-E(x))2 p(x-E(x)) Good 30% $40 $12$16$256 77 For each of the following probability distributions, calculate the expected value and standard deviation: a. 0000004694 00000 n The expected value E(X) is deﬁned by E(X) = X x∈Ω xm(x) , provided this sum converges absolutely.

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