A First Course In Probability, Global Edition
10th Edition
ISBN: 9781292269207
Author: Ross, Sheldon
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Question
Chapter 6, Problem 6.49P
To determine
To Compute: The density
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
A student has 3 volumes of short stories and 2 novels on a bookshelf. She selects 3 books at random to take home over vacation. A random variable X is defined to be the number of novels selected. Find the values assumed by X and its density function.
There are two components in an electronic system. Let X and Y denote the lengths of life in years. The
joint density function of these variables
f(x, y) = {ye-&+x)
,x 2 0, y
Determine
P(x>3)
What is the probability that both components will exceed 3 years?
An electronic system has one each of two different types of components in joint oper-
ation. Let Y₁ and Y₂ denote the random lengths of life, in hundreds of hours, of the
components of type I and type II, respectively. The joint density function is given by:
f(y₁, y2)
=
[(1/8)y₁e¯(v₁+y²)/²_if y₁ > 0, y2 > 0
elsewhere
0
(a) Verify that f(y₁, y2) is a probability density function.
(b) Find P(Y₁ > 1, Y2₂ > 1).
(c) Find the probability that a component of type II will have a life length in excess
of 200 hours.
(d) Is fy₁,Y₂ (y₁, y2) = fy₁ (₁) fy₂ (Y2)?
(e) One way to measure the relative efficiency of the two components is to compute
the ratio Y₂/Y₁. Find E(Y₂/Y₁).
Chapter 6 Solutions
A First Course In Probability, Global Edition
Ch. 6 - Two fair dice are rolled. Find the joint...Ch. 6 - Suppose that 3 balls are chosen without...Ch. 6 - In Problem 8 t, suppose that the white balls are...Ch. 6 - Repeat Problem 6.2 when the ball selected is...Ch. 6 - Repeat Problem 6.3a when the ball selected is...Ch. 6 - The severity of a certain cancer is designated by...Ch. 6 - Consider a sequence of independent Bernoulli...Ch. 6 - Prob. 6.8PCh. 6 - The joint probability density function of X and Y...Ch. 6 - Prob. 6.10P
Ch. 6 - In Example Id, verify that f(x,y)=2exe2y,0x,0y, is...Ch. 6 - The number of people who enter a drugstore in a...Ch. 6 - A man and a woman agree to meet at a certain...Ch. 6 - An ambulance travels back and forth at a constant...Ch. 6 - The random vector (X,Y) is said to be uniformly...Ch. 6 - Suppose that n points are independently chosen at...Ch. 6 - Prob. 6.17PCh. 6 - Let X1 and X2 be independent binomial random...Ch. 6 - Show that f(x,y)=1x, 0yx1 is a joint density...Ch. 6 - Prob. 6.20PCh. 6 - Let f(x,y)=24xy0x1,0y1,0x+y1 and let it equal 0...Ch. 6 - The joint density function of X and Y is...Ch. 6 - Prob. 6.23PCh. 6 - Consider independent trials, each of which results...Ch. 6 - Suppose that 106 people arrive at a service...Ch. 6 - Prob. 6.26PCh. 6 - Prob. 6.27PCh. 6 - The time that it takes to service a car is an...Ch. 6 - The gross daily sales at a certain restaurant are...Ch. 6 - Jills bowling scores are approximately normally...Ch. 6 - According to the U.S. National Center for Health...Ch. 6 - Monthly sales are independent normal random...Ch. 6 - Let X1 and X2 be independent normal random...Ch. 6 - Prob. 6.34PCh. 6 - Teams 1, 2, 3, 4 are all scheduled to play each of...Ch. 6 - Let X1,...,X10 be independent with the same...Ch. 6 - The expected number of typographical errors on a...Ch. 6 - The monthly worldwide average number of airplane...Ch. 6 - In Problem 6.4, calculate the conditional...Ch. 6 - In Problem 6.3 calculate the conditional...Ch. 6 - Prob. 6.41PCh. 6 - Prob. 6.42PCh. 6 - Prob. 6.43PCh. 6 - The joint probability mass function of X and Y is...Ch. 6 - Prob. 6.45PCh. 6 - Prob. 6.46PCh. 6 - An insurance company supposes that each person has...Ch. 6 - If X1,X2,X3 are independent random variables that...Ch. 6 - Prob. 6.49PCh. 6 - If 3 trucks break down at points randomly...Ch. 6 - Consider a sample of size 5 from a uniform...Ch. 6 - Prob. 6.52PCh. 6 - Let X(1),X(2),...,X(n) be the order statistics of...Ch. 6 - Let Z1 and Z2 be independent standard normal...Ch. 6 - Derive the distribution of the range of a sample...Ch. 6 - Let X and Y denote the coordinates of a point...Ch. 6 - Prob. 6.57PCh. 6 - Prob. 6.58PCh. 6 - Prob. 6.59PCh. 6 - Prob. 6.60PCh. 6 - Repeat Problem 6.60 when X and Y are independent...Ch. 6 - Prob. 6.62PCh. 6 - Prob. 6.63PCh. 6 - In Example 8b, let Yk+1=n+1i=1kYi. Show that...Ch. 6 - Consider an urn containing n balls numbered 1.. .....Ch. 6 - Suppose X,Y have a joint distribution function...Ch. 6 - Prob. 6.2TECh. 6 - Prob. 6.3TECh. 6 - Solve Buffons needle problem when LD.Ch. 6 - If X and Y are independent continuous positive...Ch. 6 - Prob. 6.6TECh. 6 - Prob. 6.7TECh. 6 - Let X and Y be independent continuous random...Ch. 6 - Let X1,...,Xn be independent exponential random...Ch. 6 - The lifetimes of batteries are independent...Ch. 6 - Prob. 6.11TECh. 6 - Show that the jointly continuous (discrete) random...Ch. 6 - In Example 5e t, we computed the conditional...Ch. 6 - Suppose that X and Y are independent geometric...Ch. 6 - Consider a sequence of independent trials, with...Ch. 6 - If X and Y are independent binomial random...Ch. 6 - Suppose that Xi,i=1,2,3 are independent Poisson...Ch. 6 - Prob. 6.18TECh. 6 - Let X1,X2,X3 be independent and identically...Ch. 6 - Prob. 6.20TECh. 6 - Suppose that W, the amount of moisture in the air...Ch. 6 - Let W be a gamma random variable with parameters...Ch. 6 - A rectangular array of mn numbers arranged in n...Ch. 6 - If X is exponential with rate , find...Ch. 6 - Suppose thatF(x) is a cumulative distribution...Ch. 6 - Show that if n people are distributed at random...Ch. 6 - Suppose that X1,...,Xn are independent exponential...Ch. 6 - Establish Equation (6.2) by differentiating...Ch. 6 - Show that the median of a sample of size 2n+1 from...Ch. 6 - Prob. 6.30TECh. 6 - Compute the density of the range of a sample of...Ch. 6 - Let X(1)X(2)...X(n) be the ordered values of n...Ch. 6 - Let X1,...,Xn be a set of independent and...Ch. 6 - Let X1,....Xn, be independent and identically...Ch. 6 - Prob. 6.35TECh. 6 - Prob. 6.36TECh. 6 - Suppose that (X,Y) has a bivariate normal...Ch. 6 - Suppose that X has a beta distribution with...Ch. 6 - 6.39. Consider an experiment with n possible...Ch. 6 - Prob. 6.40TECh. 6 - Prob. 6.41TECh. 6 - Each throw of an unfair die lands on each of the...Ch. 6 - The joint probability mass function of the random...Ch. 6 - Prob. 6.3STPECh. 6 - Let r=r1+...+rk, where all ri are positive...Ch. 6 - Suppose that X, Y, and Z are independent random...Ch. 6 - Let X and Y be continuous random variables with...Ch. 6 - The joint density function of X and Y...Ch. 6 - Consider two components and three types of shocks....Ch. 6 - Consider a directory of classified advertisements...Ch. 6 - The random parts of the algorithm in Self-Test...Ch. 6 - Prob. 6.11STPECh. 6 - The accompanying dartboard is a square whose sides...Ch. 6 - A model proposed for NBA basketball supposes that...Ch. 6 - Let N be a geometric random variable with...Ch. 6 - Prob. 6.15STPECh. 6 - You and three other people are to place bids for...Ch. 6 - Find the probability that X1,X2,...,Xn is a...Ch. 6 - 6.18. Let 4VH and Y, be independent random...Ch. 6 - Let Z1,Z2.....Zn be independent standard normal...Ch. 6 - Let X1,X2,... be a sequence of independent and...Ch. 6 - Prove the identity P{Xs,Yt}=P{Xs}+P{Yt}+P{Xs,Yt}1...Ch. 6 - In Example 1c, find P(Xr=i,Ys=j) when ji.Ch. 6 - A Pareto random variable X with parameters a0,0...Ch. 6 - Prob. 6.24STPECh. 6 - Prob. 6.25STPECh. 6 - Let X1,...,Xn, be independent nonnegative integer...
Knowledge Booster
Similar questions
- The random vector (X, Y ) has the following joint probability density function:f(X,Y )(x, y) = 4xye^−(x2+y2), x > 0, y > 0,0 , otherwiseLet Z =√(X^2 + Y ^2) . Find the probability density of the random variable Z.arrow_forwardThe time until a chemical reaction is complete (in milliseconds) has the density function -x/100 f(x) = for x > 0. 100 a) Determine the mean and variance of X. b) P(X > 200).arrow_forwardFind S 2t?e*dtarrow_forward
- The life lengths of two transistors in an electronic circuit is a random vector (X; Y ) where X is the life length of transistor 1 and Y is the life length of transistor 2. The joint probability density function of (X; Y ) is given by | 2e-(x+2y) x 2 0, y 20 fx,y(x,y) = fx.MX.v) else Then the probability that the first transistor last for at least half hour given that the second one lasts at least half hour equals Select one: a. 0.3669 b. 0.3935 c. 0.7772 d. 0.6318 e. 0.606arrow_forwardThe number X is chosen at random between 0 and 1. determine the density functions of the random variables V = X/(1 - X) and W = X(1 - X)arrow_forwardA system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months) by f(x) = Cxe-x/2 x > 0 x ≤0 least 5 months? Yanıt: what is the probability that the system functions for atarrow_forward
- A system consisting of one original unit plus a spare can function for a random amount of time X. If the density of X is given (in units of months) by f(x) = Cxe-x/2x>0 x ≤0 Yanıt: 0 least 5 months? what is the probability that the system functions for at 12arrow_forwardLet (X,Y)' have density for r, y > 0, f (x, y) = (1+x)² (1+xy) > 0, otherwise. Show that X and X Y are independent, equidistriduted random variables and determine their distribution.arrow_forwardAn instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the chemical is first determinedto be present. At very low concentrations, however, the chemical may fail tobe detected even if it is present. This phenomenon is modeled by assuming thatif the concentration is x, the chemical is detected with probability R(x). Let Ydenote the concentration of a chemical in a soil in which it has been determinedto be present. Show that the density function of Y isg(y) = R(y) f (y)/An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the…arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:9780134753119
Author:Sheldon Ross
Publisher:PEARSON