4. (a) Consider a nonnegative, integer valued random variable Y , and a random sample X1,., Xn.+ Xy) in terms of E(X1)Assume Y and all the X;'s are independent. Find E(X1+and E(Y)....(b) Suppose each person who logs onto Amazon.com on Christmas Eve is expected to spend$80. One hundred people are randomly chosen to see how much is spent by them. Eachof them visits Amazon.com with probability 60%. How much money do we expect thisgroup of people to spend at Amazon.com?

Expectation gives the most likely value of a random variable.

Answered: 4. (a) Consider a nonnegative, integer…

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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Let P and Q be two independent Random variables with Var(P) = 5 and Var(Q) = 4. Find Var(7P+4Q-1).
Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.
7) Suppose that X and Y are two independent measurements. We can treat these measurements as random variables, where X has the mean of 1 and standard deviation of 2, and Y has the mean of 2 and the standard deviation of 3. For the transformations below, find the mean, variance, and the standard deviation. a) X+Y b) X-Y c) 2X+Y d) X-2Y+1
B5. Let X₁, X₂, ..., Xn be IID random variable with common expectation µ and common variance o², and let X = (X₁ + + X₂)/n be the mean of these random variables. We will be considering the random variable S² given by (a) By writing or otherwise, show that S² (b) Hence or otherwise, show that n S² = (x₁ - x)². = Ĺ(X₂ i=1 X₁ X = (X₁-μ) - (x-μ) = Σ(X; -μ)² - n(X - μ)². i=1 ES² = (n-1)0². You may use facts about X from the notes provided you state them clearly. (You may find it helpful to recognise some expectations as definitional formulas for variances, where appropriate.) (c) At the beginning of this module, we defined the sample variance of the values x₁, x2,...,xn to be S = 1 n-1 n i=1 ((x₁ - x)². Explain one reason why we might consider it appropriate to use 1/(n-1) as the factor at the beginning of this expression, rather than simply 1/n. B6. (New) Roughly how many times should I toss a coin for there to be a 95% chance that between 49% and 510/ of my nain toon land Honda?
Suppose X is a random variable such that E(X) = 90 and Var(X) = 7. Compute thefollowing: E(4X + 12) Var(−X) SD(4X)E(X^2) E(X + 3)^2
The random variable X takes value O with probability, and value 1 with the remaining probability. The random variable Y takes value 100 with probability and value 101 with the remaining probability. What is the relation between the variance of X and the variance of Y? Var[X] Var[Y] "
2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
Let x1, x2, ..., n represent a random sample from a distribution with mean E(x) and variance Var(x). Show that Cov(x, x₁ - x) = 0.

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