Given that a random variable X can be assumed to have the values 1,2, 3, withprobabilities p1, P2, P3,. where P is the probability that X equals k with k = 1, 2, 3, ..Suppose that Px > 0 and Eı Pk = 1, while the expected value of X, denoted by E(X), isthe number E, kpz provided the series converges.1.a) Differentiate - to obtain a summation series for(1-x)2given that:1-x11+ x +x2 + ... + x"-1+ x"1- xb) Given pk = 2-k, show that E1 Pk =1 and find E(X), if it exists.

a) From the given information, 11-x=1+x+x2+x2+....+xn-1+xn Differentiate the above sequence,…

Answered: Given that a random variable X can be…

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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3. Let X₁,..., X36 be a random sample from a uniform distribution on interval (-8, 7). Find the (approximate) conditional probability 36 36 P| ΣX; > 0 | ΣX₁ < 20 ) i=1 i=1
Suppose that two teams play a series of games that ends when one of them has won i games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when i = 2. Also, show that this number is maximized when p = 1/2.
Every week a company randomly selects a number n of its workers for health checks. The proportion of its workers who are over 40 years old is p. Time left 1:27:3 2 Let X represent the number of over-40 workers selected in a particular week so that X ~ Bin(n,p). Suppose Xhas the probability distribution below: x P(X = x) 0 ? 1 0.36015 2 0.30870 3 0.13230 4 0.02835 5 0.00243 a. State the value of n: b. Find the value of p. Give answer to 1 decimal place: The following year, p does not change but the company decides to select 8 workers each week for health checks. c. How many combinations of 5 over-40 workers are possible within the set of 8 workers selected for the health checks?
1. Let X, 2 Poisson(A,), for i = 1,2. (a) Show that X1 + X2~ Poisson(A1 + Ag). (b) Show that X| X + Y is binomial with probability of success equal to (c) What is the distribution of X | X +Y?
Students arrive at Panda Express in the student center in accordance with a Poisson processwith rate λ. However, if the the student arriving finds n students already in line (including all who arewaiting and who are currently being served), then the student will enter the line with probability αn.Assume that there is only one person working at Panda Express and the time spent with each studentfollows an exponential distribution with rate μ. Set up this process as a birth and death process anddetermine the birth and death rates. Draw the rate diagram (state transition diagram).
3. For a time series with linear trend i.c. Y = Bo + B₁t + Ct, verify the least squares estimators for ₁ on slide 12.
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Two resistors, with resistances R1 and R2, are connected in series. R1 is normally distributed with mean 100 Ω and standard deviation 5 Ω,and R2 is normally distributed with mean 120 Ω and standard deviation 10 Ω. a) What is the probability that R2 > R1? b) What is the probability that R2 exceeds R1 by more than 30 Ω.
A and B agree to meet at a certain place between 1 PM and 2 PM. Suppose they arrive at the meeting place independently and randomly during the hour. Find the distribution of the length of time that A waits for B. (If B arrives before A, define A's waiting time as 0.)
A time series consists of observations x₁, x2,...,x, taken on a random variable at equally- spaced points of time. If, of three consecutive observations x,-1, x, and x+1, x, is the smallest or largest of the three, then there is said to be a turning-point in the series at time. Let S be the total number of turning-points in the series of n observations. (a) For the case n = 4, suppose that x₁, x2, x3 and x4 take the values 1, 2, 3 and 4 in some order. Write down all the permutations of these four numbers, and for each permutation write down the value which S would take if x₁ were the first number, x2 the second, and so on. If all permutations are equally likely, what is the probability function of S? (b) Now suppose that x1, x2,..., X are independent observations, all from the same continuous probability distribution. Show that the expected value of S is (n -2).
Assume that a time-series plot takes the form of that shown in the following graph: 80 60 60 40 40 20 20 * -20 ** * -40 0 2 4 6 8 10 12 Given this plot, which of the following models would likely give the best fit? O ^ = bo + bit + by + bị t lý = bo + bit + bit ŷ = bob₁ + b₁t On =bo+ bạt + biết biết ba
b) Sketch the probability mass function and the cumulative distribution function of X (c) Consider a random sample of 6 parcels. If Y is the number of parcels taking longer than 1 day to arrive, write down the distribution of Y and its mean and variance.

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