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- Suppose the probability of erroneously transmitting a single digit is P=0.03. Compute the probability of transmitting a 4-bit code word with (a) at most one error, and (b) exactly four errors.The event that exactly one of the events A or B occurs can be represented as (A intersection B') U (A' intersection B). Show that P[(A intersection B') U (A' intersection B)] = P(A)+P(B) - 2P(A intersection B)Suppose X is random variable whose p.d.f. is f2)=(2x-x²), 0, OSxs2 Suppose X is random variable whose p.d.f. is f(x)={4 elsewhere Find the mode , if it exists.
- Inside of a jar are four balls that are labeled 1, 2, 3, and 4. You randomly choose three balls from the jar (w/ replacement). Let X = smallest of the three numbers on the balls selected (i) What is the pmf for X? (ii) What is the CDF for X? (iii) What is P(X<3)?(3) Five balls are randomly selected without replacement from an urn with 11 black and 10 white balls. Let X be the number of black balls selected. Compute Var (X).= 4. Consider 3 events A, B and C with Pr(A) = a, Pr(B) = 3a, Pr(C) = 5a, and Pr(An B) = Pr(ANC) = Pr(BC) = 28. Prove that ß ≤ 11. 14
- An urn contains m number of red balls and n number of white balls. Two balls are selected without replacement at random from the urn. Prove P(first ball is red | second ball is white) = m/(m+n-1)As a generalization of Example 5.3(figure), consider a test of n circuits such that each circuit is acceptable with probability p, independent of the outcome of any other test. Show that the joint PMF of X, the number of acceptable circuits, and Y, the number of acceptable circuits found before observing the first reject, is PX,Y(x,y) = ((n-y-1)C(x-y))*p^(x)*(1-p)^(n-x) For 0 ≤ y ≤ x < n p^(n) For x=y=n 0 otherwise Hint: For 0 ≤ y ≤ x < n, show that {X = x, Y = y} = A ∩ B ∩ C, where A: The first y tests are acceptable. B: Test y + 1 is a rejection. C: The remaining n − y − 1 tests yield x − y acceptable circuitsThe process for cleaning up waste in a nuclear reactor core room eliminates 85% of the waste present in the area. If there is 1.7 kg of waste in the room at the beginning of the monitoring period and 2 kg of additionalwaste are generated each week, determine a recurrence relation and initial conditions describing the amount wn, of waste in the core room at the end of week n of the monitoring period.
- Let A₁-(2, 4, 6, ....., 2i} where i = 1, 2, 3, ... ex A7 (2.4.6.8.10.12.143 1) Find A F4 {2,4,.., 2n-3} % F5 B {2,4,.., 2n-2} O UA, Il app.honorlock.com is sharing your screen. Stop sharing Hide A F6 O C {2,4,.., 2n-1} D {2,4,..., 2n} F7 PrtScn F8 Calculator Home F9 W Guidel End O F101) Consider a memoryless discrete-time source which emits a sequence of random elements X1, X2, ,Xk,··, where xk for each integer k can be any of a1, a2, a3 and a4. Assume •', that Prob(xk = a1) = 0.01, Prob(Xk = a2) 0.1 and Prob(xk a3) = 0.3. Answer the following questions: d) For each fixed integer l, we can apply Huffman's method to encode average realization of (Tnl+1, Xnl+2, · ·· , Xnl+n) where n is another integer. The average codeword length of the resulting codebook is denoted by Ln. What is the limit of Ln/n as n → 0? Is this codebook for xnl+1, Xnl+2, , X'nl+n dependent on l? Why?5. Suppose you are tossing a fair coin to bet. At each time your gain is X;. The rule is: (i) if you get heads at i-th toss, X; = 1$; (ii) if you get tails at i-th toss, X; = -1$. Let Wn = X1 + X2 + · · · + Xn with Wo = 0. Note that Wn represents your total wealth after n tosses. Suppose after 100 tosses you have 11$, that is W100 = 11. What is your expected wealth if you do another 100 tosses, that is EW200|W100 = 11]?
