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Using the Exponential Density Function In Exercises 33-36, find the required probabilities using the exponential probability density function
Waiting Time The waiting time t (in minutes) for service at the checkout at a grocery store is exponentially distributed with
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Chapter 9 Solutions
Calculus: An Applied Approach (MindTap Course List)
- a) Find the conditional probability density function under the condition A = {X> 1/8}.b) Find the domain of the function.arrow_forwardPROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE E[cg(X,Y)] = cE[g(X,Y)]arrow_forwardDo you dislike waiting in line? A supermarket chain has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system, which allows the supermarket to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 28 seconds. (a)Assume that supermarket waiting times are exponentially distributed. Show the probability density function of waiting time at the supermarket. f(x) = x ≥ 0 elsewhere (b) What is the probability that a customer will have to wait between 30 and 45 seconds? (Round your answer to four decimal places.) (c) What is the probability that a customer will have to wait more than 2 minutes? (Round your answer to four decimal places.)arrow_forward
- Using the uniform probability density function shown in Figure, find the probability that the random variable X is between 1.0 and 1.9.arrow_forwardProof Let X and Y be independent integrable random variables on a probability space and f be a nonnegative convex function. Show that E[f(X +Y)] ≥ E[f(X +EY)].arrow_forwardPROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE E[g(X,Y) + h(X,Y)] = E[g(X,Y)] + E[h(X,Y)]arrow_forward
- PROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE If X and Y are independent, then E[XY] = E[X] E[Y]arrow_forwarda) Find the conditional probability density function under the condition A = {X> 1/8}.b) Find the domain of the function.c) Find the conditional expected value of X.arrow_forwardDo you dislike waiting in line? A supermarket chain has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system, which allows the supermarket to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 23 seconds. (a) Assume that supermarket waiting times are exponentially distributed. Show the probability density function of waiting time at the supermarket. f(x) = , x ≥ 0 , elsewhere (b) What is the probability that a customer will have to wait between 30 and 45 seconds? (Round your answer to four decimal places.) (c) What is the probability that a customer will have to wait more than 2 minutes? (Round your answer to four decimal places.)arrow_forward
- Proof. Let X be a random variable and let g(x) be a non-negative function. Then for r>0, P [g(X) ≥ r] ≤ Eg(X)/rarrow_forwardPROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE If X = Y, then Cov(X,Y) = Var(Y)arrow_forwarduse a graphing utility to graph the function. Determine whether the function f represents a probability density function over the given interval. If f is not a probability density function, identity the conditions that is (are) not satisfied f(x)= 1/8 (0,8)arrow_forward
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