Let X be a Poisson random variable with E[X] = ln2. Calculate E[cosπX]
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1. Let X be a Poisson random variable with E[X] = ln2. Calculate E[cosπX].
2. The number of home runs in a baseball game is assumed to have a Poisson distribution with a mean of 3. As a promotion, Mall A pledges to donate 10,000 dollars to charity for each home run hit up to a maximum of 3. Find the expected amount that the company will donate. Mall B also X dollars for each home run over 3 hits during the game, and X is chosen so that the Mall B's expected donation is the same as the Mall A's. Find X.
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- Let i_t denote the effective annual return achieved on an equity fund achieved between time (t -1) and time t. Annual log-returns on the fund, denoted by In(1 + i_t) , are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and o = 14%.An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%. Using only formulas, no tablesConsider two securities, the first having μ1 = 1 and σ1 = 0.1, and the secondhaving μ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated,with ρ = −0.8. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve in R.At 15:00 it is the end of the school day, and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average, 24 students leave the school every minute. (e) There are 200 days in a school year. Given that Y denotes the number of days in the year that at least 700 students leave before 15:30, find (ii) P(Y > 150).
- 1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.Suppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.
- In the daily production of a certain kind of rope, the number of defects per foot given by Y is assumed to have a Poisson distribution with mean ? = 4. The profit per foot when the rope is sold is given by X, where X = 70 − 3Y − Y2. Find the expected profit per foot.The lifetime of a certain type of TV remote control is given by Y . Suppose Y has approximately exponential distribution with mean 8 years. a) Find the probability that a remote control of this type will last more than 15 years. b) Find the probability that of eight such remote controls at least one will last more than 15 years. c) What should the warranty period for these remote controls be if the manufacturer wants 85% of the remote controls to last beyond the warranty period? d) What is the moment generating function of Y .1i. Suppose that a structure can withstand a flood with a peak discharge no greater than 1,950 m3/s and it has been designed to have an economic life span of 100 years. Determine the risk of failure assuming the Extreme Value Distribution when the mean and standard deviation of the annual flood series are 410m3/s and 280 m3/s, respectively. 1ii. What is the probability that at least one flood of ARI 50y (T=50y) will occur during the 30 year design life of a flood control project? 1iii. Consider a small, temporary (3 year design life) flood mitigation dam designed to contain a 20 year flood event. What is the risk that it will be overtopped at least once in the design life.
- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Let X = the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with λ = 1 Compute P(X≤4) and P(2≤X≤5)