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Problem 2: 15 points (= 5 + 5 + 5] Assume that a random variable (Y) follows the exponential distribution with density f(y) = 2e-2 for y>0 and f (y) = 0 elsewhere. Given Y = y, a random variable N is Poisson distributed with rate = y. 1. Determine marginal distribution of N. 2. Evaluate expectation of N. 3. Derive variance of N.

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Related textbook solutions

Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

Related questions

Q5. Let (Ω, F , P) be a probability space. Let λ > 0 be some constant.

  1. (a)  Give the definition of a random variable X on R ∪ {−∞, +∞}.

  2. (b)  Define the distribution function FX for the RV X.

[6 marks]

[1 marks]

[1 marks] (c) Suppose that Y is a RV on R ∪ {−∞, +∞} with distribution function given by

􏰄3 −1e−λy (y≥0)

[6 marks]

page1image62001664 page1image62003968

FY (y) = 4 2 1

(y < 0).
For any a < b, determine P(Y > a), and P(Y ∈ (a, b]). [3 marks]

  1. (d)  By noting {Y < b} = ∪n∈N{Y ≤ b − 1 } and using Continuity of Probability, n

    find P(Y < b). Similarly, find P(Y < ∞) and P(Y = −∞). [4 marks]

  2. (e)  Is Y either a discrete RV, or a continuous RV, or a mixture of both? Explain

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Problem 1 (2 points). Every month a clothing store conducts an inventory and calculates the losses due to theft. The store would like to reduce these losses and is considering two methods to do so: the first is to hire a security guard and the second is to install cameras. The manager of the store has conducted the experiment: During the first 6 months he hired a security guard (method 1) and during the next 6 months he installed cameras (method 2). The monthly losses were reported as follows:

Security guard 350 289 407 398 470 259 Cameras 488 301 276 380 421 425

a/ Test with a=0.05 to determine whether there is enough evidence to conclude that expected monthly loss when installing cameras differs from 382. b/Since the cameras are cheaper than the guard, the manager would prefer to install the cameras unless there was enough evidence to infer that the guard was better. Conduct a hypothesis testing to advise what the manager should do? (Assuming equal variances and using a= 0.05)

Problem 2 (2 points). A supermarket chain performed a survey to help determine desirable locations for its new stores. The management of the chain wanted to know whether a linear relationship exists between weekly take home pay (s) and weekly food expenditures (y). A random sample of eight households produced the data shown in the following table Household 1 3 7 8 2 4 5 6 550 450 560 340 510 710 470 660

170 100 160 80 120 190 140 140 Y

a/ Find the least square regression line.

b/Perform the test Ho: Bi=0, H4: B10 (using a = 0.05) to determine whether there is enough evidence to infer that a linear relationship exists between weekly take home pay (s) and weekly food expenditures (y).

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