Problem 3i.e.Let W, be a negative-binomial variable with parameter p € (0,1),P{W, = k} =p' (1-p)-rifk = r,r+1,...,else.Let furthermore (ilieN be a sequence of independent Bernoulli variables with parameter p. Re-call that W, has the same distribution asY₁ =kX, := min {k € N₁: [{i=r},i=1i.e. the number of independent Bernoulli trials with parameter p that we have to observe untilwe see the rth success.Define the random variables Yo, Y₁,..., Y, by0min {k EN: EY+-+Y₁-1+k=1}for i = 0,for i = 1,...,r.(a) Compute P{Y;=k | Y₁ = ₁,...,Y₁-1 = li-1} for positive integers k, l,...,li-1.(b) It is possible to show that Y₁, Y2,... are independent. Give a heuristic explanation for thisfact.(c) Show by induction over rEN, that X, Y₁ +...+Yr.=

We have Wr negative binomial random variable with parameter r and p and PMF given by…

Answered: Problem 3 i.e. Let W, be a…

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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Problem 4. Let Φ be the standard distribution function. Show that Φ(B(t)) / √ T − t is a martingale for 0 ≤ t < T using Itô-formula. please use definition and reasons while solving the problem.
Problems 5 and 6 refer to the discrete random variables X and Y whose joint distribution is given in the following table, so P(X = 1 and Y = -1) = 1/4, P(X = 1 and Y = 1) = 0, etc. Problem 5: Compute the marginal distributions of X and Y, and use these to compute E(X), E(Y), Var(X), and V ar(Y). Problem 6: Compute Cov(X, Y) and the correlation ρ for the random variables X and Y. Are X and Y independent? Y= -1 Y =0 Y =1 X =1 1/4 1/8 0 X =2 1/16 1/16 1/8 X =3 1/16 1/16 1/4
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1..Consider that a game involves the spinning of a dial which is not fair. As the dial is not fair, after spinning it is more likely to point in any particular direction than another. Suppose that the movement of a dial, X, can be modelled by the following probabilityfunction f(x) = A sin x; 0 ≤ x ≤ π. (i) Determine the value of A so that f(x) is a pdf (ii) Calculate E(X) and Var(X).
QUESTION 2 Delta Airlines quotes a flight time of 4 hours, 3 minutes for a particular flight. Suppose we believe that actual flight times are uniformly distributed between 4 hours and 4 hours, 12 minutes. (a) Show the graph of the probability density function for flight time. The graph has a shaded area. The horizontal axis is labeled: x with the title: Flight Time in Minutes and has tickmarks labeled: 234, 240, 246, 252. The vertical axis is labeled: f(x), and has tickmarks labeled: 1/12, 1/6, 1/4. The shaded area is the region bounded by the horizontal axis and the following line segments. A 1/12 unit long vertical line segment begins at 240 on the horizontal axis. A 1/12 unit long vertical line segment begins at 252 on the horizontal axis. A 12 unit long horizontal line segment begins at the tickmark labeled 240 on the horizontal axis and the tickmark labeled 1/12 on the vertical axis. This line segment connects the two vertical line segments. The graph has a shaded…
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Question 18 According to a University Center for Logistics Management, 8% of all merchandise sold in the United States gets returned. A Seattle department store sampled 90 items sold in January and found that 10 of the items were returned.If you have the following null and alternative hypotheses for a test you are running:H0:p=0.08Ha:p>0.08Calculate the test statistic, rounded to 3 decimal places Z=
Q1- What is the probability that the number of heads and are equal it a lait coin is tossed, a)15 times b)20 times Q2- Suppose the joint pdf of X and Y is given by: F xy (x y)={ ex +3y, if 0<x< 1 and 0<y<1 0 otherwise find E(x) . Q2- Suppose X is a random variable with E(X) = 5 and Var(X) = 2. What is E(X)?
QUESTION 11 An educator estimates that the dropout rate for seniors at high schools in Colorado is 15%. Last year in a random sample of 300 Colorado seniors, 34 withdrew from school. At α = 0.10 level of significance, is there enough evidence to reject the educator’s claim? Yes, because the p-value is 0.075 Yes, because the z-statistic is 1.6 No, because the p-value is 0.038 Yes, because the p-value is larger than 0.15
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