Question: If X 1 , . . ., X n are independent exponentially distributed random variables, each with parameter , prove that the random variable whose

If X₁, . . ., X_n are independent exponentially distributed random variables, each with parameter λ, prove that the random variable whose value is given by the minimum of {X₁, . . ., X_n} also has an exponential distribution, with parameter nλ. In particular, if a complex piece of machinery consists of six parts, each of which has an exponential distribution of time to failure with mean 2000 hours, and if the machine fails as soon as any of its parts fail, find the probability that the time to failure exceeds 300 hours.

Step by Step Solution

★★★★★

3.36 Rating (149 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Each exponential random variable has pdf and distribu... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Modern Engineering Mathematics Questions!

The Supreme Court is seen as the ultimate judicial branch. Explain why State Courts are not as commonly recognized for their policy making activities as the Federal Courts, using a specific example

A manufacturing system consists of five workstations, and at present stations 1, 2, 5 consists of 3, 2, 4, 3, and 1 identical machine(s), respectively, as shown. In effect, the system is a network...

Provide a discussion of the external context within which the 2024 Sydney Lunar New Year Dragon Boat Festival occurs in risk management. legal aspect please.

1. (10 points) On any day, the number of accidents on a highway has a Poisson distribution with parameter A. The parameter A varies from day to day and is itself a random variable. Suppose that A is...

where bc is the \"round down\" operator (that is, for example, b2.5c = 2, b 2.5c = 3, b 3c = 3). Find the probability mass function of Y . (Hint: For Y to take value k, what values should X take?) c)...

=+1. Let U, V , and W be independent exponentially distributed random variables with intensities , , and , respectively. Consider the random variables X = min{U, W} and Y = min{V,W}. Demonstrate that...

Show the workings 1.5 Exercise 1.6 Xi's are independent, exponentially distributed random variables with a mean value of 1/a, a > 0, i = 1,2. ..., n. Calculate P(X 42.31 (E) The range is empty. 12....

solve this problem im stuck in the question QUESTION 3 (10 marks) Let (Xi)=1, n E N be independent identically distributed random variables, each following a density, parameterized by the parameter 0...

=+4. Let X1, Y1, X2, Y2,... be independent exponentially distributed random variables with mean 1. Define Nx = min{n : X1 + + Xn > x} Ny = min{n : Y1 + + Yn > y}. Demonstrate that Pr(Nx Ny)=1 ey ...

Let X and Y be independent exponentially distributed random variables with identical probability density functions, i.e., fx(x) = de-A for a > 0, fy (y) = de-v for y > 0, for some A > 0, and zero...

A female employee who is a single parent of two children is being sexually harassed by her supervisor in the form of unwanted physical touching. She is worried about reporting her supervisor because...

During the month of January 20-2, TJs Specialty Shop engaged in the following transactions: Jan. 1 Sold merchandise on account to Anne Clark, $3,000, plus tax of $150. Sale No. 643. 2 Issued Check...

1. After reading this chapter and understanding how lines form and can be mitigated, do you think companies should try to eliminate lines in every case? What should a company consider when deciding...

The Internet is increasing price transparency across international markets. Fully evaluate the problems and opportunities this brings to the company trying to build a global competitive advantage.

For the tank shown in Fig. 3.21, determine the depth of the oil h if the reading of the bottom pressure gage is 35.5 psig, the top of the tank is sealed, and the top gage reads 30.0 psig. Air Oil (sg...

For the tank in Fig. 3.22, compute the depth of the oil if the depth of the water is 2.80 m and the gage at the bottom of the tank reads 52.3 kPa(gage). Vent Oil (sg = 0.86) h2 Water

For the tank in Fig. 3.22, compute the depth of the water if the depth of the oil is 6.90 m and the gage at the bottom of the tank reads 125.3 kPa(gage). Vent Oil (sg = 0.86) h2 Water

Coasty Inc. is considering automating some parts of the existing assembly line. The purchase and installation of the required equipment cost 400,000. The cost will be appreciated straight-line to...

In the SQL server format, with books of authors and the variables au_lname and au_fname Display the names, last and first, of all authors who have ls in the third and forth positions in their last...

Billys Bar has a beta of 1.2 and currently sell for $44.01 per share. It will pay an annual dividend of $2.63 per share next year. The company has a growth rate of 1.7% annually. The return of market...