A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 10, Problem 10.6P
Give a method for simulating a random variable having failure rate
a.
b.
c.
d.
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A random variable Y has a uniform distribution over the interval (?1, ?2). Derive the variance of Y.
Find E(Y)2 in terms of (?1, ?2).
E(Y)2 =
Find E(Y2) in terms of (?1, ?2).
E(Y2) =
Find V(Y) in terms of (?1, ?2).
V(Y) =
Consider the following
simple linear regression.
iid
Y = Po + B₁X + e, e~ N(0,02), i=1,..., n.
2. Let X be a random variable with pdf fx(x), and Y = X².
%3D
(a) Find fx(x|X > 0)
(b) Find fy(y|X > 0)
Chapter 10 Solutions
A First Course in Probability
Ch. 10 - The following algorithm will generate a random...Ch. 10 - Prob. 10.2PCh. 10 - Give a technique for simulating a random variable...Ch. 10 - Present a method for simulating a random variable...Ch. 10 - Use the inverse transformation method to present...Ch. 10 - Give a method for simulating a random variable...Ch. 10 - Let F be the distribution functionF(x)=xn0x1 a....Ch. 10 - Prob. 10.8PCh. 10 - Suppose we have a method for simulating random...Ch. 10 - Prob. 10.10P
Ch. 10 - Use the rejection method with g(x)=1,0x1, to...Ch. 10 - Prob. 10.12PCh. 10 - Prob. 10.13PCh. 10 - Prob. 10.14PCh. 10 - Prob. 10.15PCh. 10 - Let X be a random variable on (0, 1) whose density...Ch. 10 - Prob. 10.1STPECh. 10 - Prob. 10.2STPECh. 10 - Prob. 10.3STPECh. 10 - If X is a normal random variable with mean and...Ch. 10 - Prob. 10.5STPE
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.arrow_forwardFind the equation of the regression line for the following data set. x 1 2 3 y 0 3 4arrow_forwardConsider the following simple linear regression. iid Y = Po + B₁X + e, e~ N(0,02), i 1,..., n.arrow_forward
- A firm's revenue R is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of effort e E [0, ). The choice of effort affects revenue so that: E(R|e) = e and Var(R|e) = 1 %3D where E(R|e) and V ar(R|e) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level e. The employer cannot observe the level of effort exerted by the employee. The employer wants to design a wage contract w based on the revenue and considers only contracts of the form: w-α+ βR and so the employee is guaranteed a payment a and then a bonus payment ßR which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w gives expected utility: Eu(w\e) = E(w\e)-eV ar(w]e) – c(e) where E(wle) and Var(wle) denote the expected value and variance of the contract, re- spectively, conditional on effort e, p is a parameter of risk aversion, and c(e) denotes the disutility of…arrow_forward2. Logistic regression and LDA. The LDA model posits the (joint) distribution of (X,Y) is Y~ Ber(π), X|Y=y~ N(µy, Σ), where € (0, 1) is the class proportion, and µo, µ₁ € Rª, and Σ € Sdxd are the means and (co)variance of the class conditionals. Show that the predictive distribution of Y in the LDA model satisfies the logistic regression model; i.e. P{Y = 1 | X = x} = 1 1+ exp(30 + 3x) where Bo and 3 are functions of π, μo, μ₁, and Σ. Hint: Recall the pdf of a N(u, E) random vector is p(x) = 1 - (2π) ¾ det(T) ³ exp(−½(x − µ)¹¶−¹(x − µ)).arrow_forward. Let Y be a continuous random variable where f(y) F(y)? (a) Y 18 Y 15 Y-3 18 Y+3 (b) (c) (d) 18 (e) None of the above = 1 18 -3 ≤ y ≤ 15. What isarrow_forward
- The p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) karrow_forwardWith an American penny, the likelihood of getting H when it is spun on edge is 0.3. If X is the random variable where X(H ) = 1, X(T ) = −1, find the expected value E(X), the variance, Var(X), and express X in its standard form.arrow_forwardLet X1, X2, . . . are independent indicator variables with different probabilities of success. That is, P(Xi = 1) = pi. Define Yn = X1 + X2 + . . . + Xn. Find E(Yn), V ar(Yn) and coefficient of variation of Ynarrow_forward
- (c) Find the expected value of Y. (d) Find the variance of Y.arrow_forwardUse what you know about order statistics to show that for the random sample of size n = 3 the median is an unbiased estimator of the parameter θ of a uniform population with α = θ − 1/2 and β = θ + 1/2.arrow_forwardTrue or False only....arrow_forward
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