Exercise 2.7. Consider an unfair coin with probability p of heads.(a) The coin has been tossed (a deterministic number) n times. Denote by Xthe number of heads among the n tosses and by Y the number of tails. Showthat X and Y are dependent.Poisson(A).(b) The coin has been tossed (a random number) N times where N -Denote by X the number of heads among the N tosses and by Y the numberof tails. Show that X and Y are independent.

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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7. Suppose that X is a real valued random variable and : R → R is continuous. Show that if X and (X) are independent, then (X) must be deterministic (i.e. there is a real number a so that P((X) = a) = 1).
Please help to solve this. Does this question use combinatorics? Since we are told that marbles are selected at random (together, without replacement). A box has 1 red and 3 blue marbles.Two marble are selected at random (together, without replacement).Let R be the number of reds we get.Let B be the number of blues we get.Compute Cov(R, B).
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Exercise 3. (Use Gaussian-elimination and show all the steps!) B and z= and w are linear combinations of x, y and z. Let x = If v and w== , determine whether v
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Advanced Math (a) Suppose X, Y, Z:2R are three discrete random variables. Prove that (Px py) Pz Px (py pz). This says that the convolution product is associative. (b) Suppose X and Y are discrete random variables with X Bin(4, ) and Y ~ Bin(5, ). Using the definition of convolution, calculate px + py. (Vandermonde's identity may be useful here.) Can Kou Gnd y quick way of deing this calculation by thinking of Y and Y ax indenendent random
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Let Xo, X₁, ... be i.i.d normal N(0, o²) random variables. Let the random variables Yo, Y₁, ... be defined as follows: Yo = Xo 3 4 Y₁ ==Y₁ +=X₁ 5 % 3 Y₂ = =Y₁ +5X₂ *** 3 4 Yn = 5 Yn-1 + 5 Xn n = 1,2,3, ...
Q6 You have two machines. Machine 1 has a lifetime T which is exponentially distributed with parameter d1, and Machine 2 has a lifetime T, which is exponentially distributed with parameter A, and is independent of T1. Machine 1 starts at time 0, and Machine 2 starts at a time T > 0. Q6(i.) Assume that T is deterministic. Compute the probability that Machine 1 is the first to fail. Q6(ii.) Answer the same question when T is exponentially distributed with parameter u and is independent of T1 and T,.
1. (Independence). (i) Recall that given events A, B,C, we have that P(AUBUC) = P(A)+ P(B)+ P(C) – P(An B) – P(AnC)– P(BnC) + P(AN BnC). (4.1) Prove that whenever events A, B,C are mutually independent, we have that P(AUBUC) = 1– P(A^)P(B^)P(C*) = 1– (1– P(A)) · (1 – P(B)) · (1 – P(C)) (and the probability in question can be calculated considerably faster than with the use of the formula (4.1)). (ii) The probability that A hits a target is 2/5, the probability that B hits it is 5/9, and the probability that C hits the target is 3/7. Use (i) to find the probability that the target will be hit if A, B, and C each shoot at the target. (iii) Suppose that the probability that a soldier firing his personal weapon hits an enemy warplane is p > 0. Show then, arguing as in (i), that the probability that the warplane is hit at least once when n > 2 soldiers shoot at it is 1- (1 — р)". (4.2) Evaluate the probability in (4.2) in the case when p = 0.0006 and n = 750, and then round your result to a…

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