A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Textbook Question
Chapter 10, Problem 10.16P
Let X be a random variable on (0, 1) whose density is f(x). Show that we can estimate
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4. (i) Let a discrete sample space be given by
N = {W1, W2, W3, W4},
and let a probability measure P on be given by
P(w1) = 0.2, P(w2) = 0.2, P(w3) = 0.5, P(wa) = 0.1.
Consider the random variables X1, X2 → R defined by
X₁(w1) = 1, X₁(w2) = 2,
X2(w1) = 2, X2 (w2) = 2,
Find the joint distribution of X1, X2.
(ii)
X1(W3) = 1, X₁(w4) = 1,
X2(W3) = 1, X2(w4) = 2.
[4 Marks]
Let Y, Z be random variables on a probability space (, F, P).
Let the random vector (Y, Z) take on values in the set [0, 1] x [0,2] and let the
joint distribution of Y, Z on [0, 1] x [0,2] be given by
1
dPy,z (y, z) ==(y²z+yz2) dy dz.
harks 12 Find the distribution Py of the random variable Y.
[8 Marks]
marks 11
3
3/4 x 1/4
1.
There are 4 balls in an urn, of which 3 balls are white and 1 ball is
black. You do the following:
draw a ball from the urn at random, note its colour, do not return the
ball to the urn;
draw a second ball, note its colour, return the ball to the urn;
finally draw a third ball and note its colour.
(i) Describe the corresponding discrete probability space
(Q, F, P).
[9 Marks]
(ii)
Consider the following event,
A: Among the first and the third balls, one ball is white, the other is black.
Write down A as a subset of the sample space and find its probability, P(A).
[2 Marks]
There are 4 balls in an urn, of which 3 balls are white and 1 ball isblack. You do the following:• draw a ball from the urn at random, note its colour, do not return theball to the urn;• draw a second ball, note its colour, return the ball to the urn;• finally draw a third ball and note its colour.(i) Describe the corresponding discrete probability space(Ω, F, P). [9 Marks](ii) Consider the following event,A: Among the first and the third balls, one ball is white, the other is black.Write down A as a subset of the sample space Ω and find its probability, P(A)
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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- Let (Ω, F, P) be a probability space and let X : Ω → R be a randomvariable whose probability density function is given by f(x) = 12 |x|e−|x| forx ∈ R.(i) Find the characteristic function of the random variable X.[8 Marks](ii) Using the result of (i), calculate the first two moments of therandom variable X, i.e., E(Xn) for n = 1, 2. [6 Marks]Total marks 16 (iii) What is the variance of X?arrow_forwardball is drawn from one of three urns depending on the outcomeof a roll of a dice. If the dice shows a 1, a ball is drawn from Urn I, whichcontains 2 black balls and 3 white balls. If the dice shows a 2 or 3, a ballis drawn from Urn II, which contains 1 black ball and 3 white balls. Ifthe dice shows a 4, 5, or 6, a ball is drawn from Urn III, which contains1 black ball and 2 white balls. (i) What is the probability to draw a black ball? [7 Marks]Hint. Use the partition rule.(ii) Assume that a black ball is drawn. What is the probabilitythat it came from Urn I? [4 Marks]Total marks 11 Hint. Use Bayes’ rulearrow_forwardLet X be a random variable taking values in (0,∞) with proba-bility density functionfX(u) = 5e^−5u, u > 0.Let Y = X2 Total marks 8 . Find the probability density function of Y .arrow_forward
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