Let X be the number of defects on the paint job of a car which follows a Poisson distribution. To repair the defects costs a fee of $100 plus $30 per defect. Let Y = the cost to repair the defects of the paint job of a car. What is the moment gener function of Y? For the instructor, this was question 8. X ek(e^t-1) X e(100+30k)(e^t-1) V e100t +k(e^30t-1) X e 100k +k(e^30t-1)
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- Find the moment generating function ME(t) for an exponential random variable with parameter (lambda) = 1. Sketch the graph of ME(t)The lifetime of a certain type of TV remote control is given by Y . Suppose Y has approximately exponential distribution with mean 8 years. a) Find the probability that a remote control of this type will last more than 15 years. b) Find the probability that of eight such remote controls at least one will last more than 15 years. c) What should the warranty period for these remote controls be if the manufacturer wants 85% of the remote controls to last beyond the warranty period? d) What is the moment generating function of Y .Suppose claim amounts at a health insurance company are independent of one another. In the first year calim amounts are modeled by a gamma random variable X with alpha=40, and beta=3. In the second year, individual claim amounts are modeled by random variable Y=1.05X+3. Let W be the average of 30 claim amounts in year two set up the equation to model the random variable W. a) Find the moment generating function of W b) Based on moment generating function of W is W also a gamma distribution? if so what are the parameters? c) Find the approximate probability that W is between 125$ and 130$.
- 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).. Consider a call option having the strike price K and exercise time t. Suppose further that thenominal interest rate is r, compounded continuously, and also that the price of the securityfollows a geometric Brownian motion with variance parameter σ^2. Derive the formula that is used to price the unique cost of the option that does not give rise to an arbitrage13) Random variables X and Y have joint pdf fXY={4xy, 0≤x≤1, 0≤y≤1fXY={4xy, 0≤x≤1, 0≤y≤1 Find Correlation and Covariance
- The probability that the Air Conditioning of a brand new car is defective is 10%. Let X be the number of cars with defective AC in a sample of size 100. (a). Find P(X < 6) exactly. (b). ) Find P(X < 6) approximately, using a Poisson approximation (c). Find P(X < 6) approximately, using a normal approximation. Note: For parts (a) and (b), you only need to provide a formal expression each. For part (c), you need to compute the numerical values using the Q-function table (Table 4.2) from the textbook.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.Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?
- A pump operates 1000 hours/year. Under a minimal repair concept, the pump failures generated a non-homogenous Poisson process having the following intensity function with t measured in operating hours. Row(t)=0.00003t^2. a) From the information given, is the rate of occurance of failure (ROCOF) increasing, decreasing or remaining constant? b) Calculate the number of expected failures of the pump over 1000 hours of operation. c) Calculate the MTBF for the 1000-hour operation. d) The repair time of the pump is best described by the following probability density function h(t)=t^2/3 for 0<_t<_3 hours. WHat is the mean time of repair, in hours? e)What is the inherent availability of the pump over the 1000 hours?Please do not give solution in image format thanku 1.Suppose that the moment generating function of a random variable X is MX(t)=exp(2e^t−2) and that of a random variable Y is MY(t)=((4/5)e^t+1/5)^16. If X and Y are independent, find each of the following. (a) P{X+Y=2}= (b) P{XY=0}= (c) E[XY]= (d) E[(X+Y)^2]= ———When two continuous variables are compared to each other in order to gain a correlation coefficient, the appropriate formula is Spearman’s Rho The Phi Coefficient Pearson’s Product Moment None of the above