13. If X has the distribution function F(x) = 0 1 12 for x < -1 for -1x < 1 for 1x <3 2 3 for 3≤x≤5 4 1 for x≥5 find (a) P(X ≤3); (b) P(X = 3); (c) P(X < 3); (d) P(X≥1); (e) P(-0.4
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- 7. Suppose that X and Y are independent random vari- ables for which Var(X)= Var(Y) =3. Find the values of (a) Var(X – Y) and (b) Var(2X – 3Y + 1). |Suppose there is a test, φ, that yields the distribution of test outcomes for good drugs and bad drugs seen in the Figure. In this case, is it possible to create a rule for accepting and rejecting drugs that yields no Type I or Type II error? If so, be sure to show this threshold rule explicitly on the graph above. If not, explain why this is impossible.2.Suppose that the return R (in dollars per share) of a stock has the uniform distribution on the interval [-3,7]. Suppose also, that each share of the stock costs $1.50. Let Y be the net return (total return minus cost) on an investment of 10 shares of the stocks. Compute E(Y).
- Q2. P(A) = 0.42, P(B|A) = 0.66 and P(B|A’) = 0.25. Find the following – P(A’), P(B’|A), P(B’|A’), P(B), P(B’), P(A|B), P(A’|B), P(A|B’), P(A’|B’)There is a chance that a bit transmitted through a digital transmission channel is received in error. Let X equal the number of bits in error in the next four bits transmitted. The possible values of X are {0,1,2,3}. Suppose that P(X=0)=0.6, P(X=1)= 0.3, P(X=2)= 0.05, P(X=3)= 0.05 What is the CDF of X at 3? i.e., what is F(3)? a). 0.6 b). 0.9 c). 0.95 d). 1 e). none of above For the random variable defined in question 12, if we know that Var(X)=0.6475, what is the standard deviation of X? a). 0.55 b). 0.6475 c). 0.419 d). 0.8 e). none of aboveplease help me
- Exercise 4.7 Calculate the system entropies where I is the binary erasure channel (BEC), introduced in §4.1, and the input probabilities of 0 and 1 are p and p. Show that this channel satisfies (4.9) and (4.10). H(BA) ≤H(B), H(AB) ≤ H(A), (4.9) (4.10)Suppose 10% of the population of WVU students has tested positive for COVID-19, W1 represents the part of the population that has tested positive and W2 represents those who tested negative. X denotes a test result that the person has tested positive for COVID-19. P(W1) = 0.10 individual tested positiveP(W2) = 0.90 Individual tested negativeP(X|W1) = 0.80 test shows the individual tested positive and is correctP(X|W2) = 0.30 test incorrectly shows the individual is positiveRandomly select an individual and preform the test. The result shows the person tested positive, what is the probability that the test is correct? Use Bayes' theorem to solve.. Suppose X ~ N(2,3). What value of x has a z-score of -0.67?
- Suppose that you enter a fantasy basketball league. Suppose that the 2022 team budget, say X, is randomly drawn from a uniform distribution on the interval [90, 210], where the unit is U.S. million dollars. In addition, suppose that after the value X = x has been observed (90 < x < 210), the 2023 team budget, say Y, is randomly drawn from a uniform distribution on the interval [x, 210]. In other words, the 2023 budget is at least as large as the 2022 budget. (a) For any given value of x (90 < x < 210), obtain E[Y|X = x]. (b) In view of part (i), obtain E[Y|X]. (c) What is the difference between E[Y|X = x] and E[Y|X]? (d) Obtain E[Y]. That is, what is the expected value of your 2023 fantasy basketball budget? (e) Golden State Warriors won the 2022 NBA finals. Their estimated 2022 payroll is about $194 million. Would your 2023 fantasy basketball budget be on average larger than Golden State Warriors' 2022 payroll? Explain briefly.Ana’s preferences are consistent with expected utility. Denote by u(x) her utility function for a monetary outcome £x, and suppose that she is strictly risk-averse. Ana currently faces the monetary lottery which pays: £A with probability pA; £B with probability pB; and £C with probability pC. Now suppose that Ana is made the following o§er. For each realization of the original lottery, another lottery will be executed according to which she wins an additional poundwith probability 1/2 and loses a pound with probability 1/2. Describe the lottery that Ana faces if she accepts the o§er by describing the new payo§s and probabilities over these payo§s. (b) Show that she rejects the o§er and explain why.6. The prosecutor's fallacy. Let G be the event that some accused person is guilty, and I the event that some testimony or evidence presented is in fact true. It has been known for lawyers to argue on the assumption that P(G|T) = P(TIG). Show that this holds if and only if P(G)=P(7).
