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- The cumulative distribution function of the continuous uniform distribution between con- stants a and b is given by F(x) = P(X ≤ x) = x-a - a x b (a) The probability density function is f(x) = F(r). Find the form of f(x). (b) Find the derivative of f(x) for x = [a, b]. Is f(x) decreasing, increasing or flat in this region? (c) Does f(x) have a single maximum in the region [a,b]? If so, what is it, or if not, why not?1. The sales tax returns of a business man is exponentially distributed with the density function given by -1Koo, f(x)= 100 x>0. What is the probability that his sales will exceed Rs.10,000 (perday), assuming that the sales tax is levied at the rate of 5% on the sales.7.1 Let X be a random variable with probability Si. 1= 1,2, 3, S(2) =10 elsewhere. Find the probability distribution of the random vari- able Y = 2X 1. 7.8 A dealer's profit, in units of $5000, on a new au- tomobile is given by Y = X², where X variable having the density function sa random S(z) =2(1 - 2), 0The lifetime, X, of a particular integrated circuit has an exponential distribution with rate of λ=0.5 per year. Thus, the density of X is: f(x,x) = 1 e-^x for 0 ≤ x ≤ ∞o, λ = 0.5. λ is what R calls rate. Hint: This is a problem involving the exponential distribution. Knowing the parameter for the distribution allows you to easily answer parts a,b,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts. Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts. d) What is the probability that X is greater than its expected value? e) What is the probability that X is > 5? f) What is the probability that X is> 10? g) What is the probability that X > 10 given that X > 5? h) What is the median of X? Please solution USING R scriptConsider times at which a parent provisions nestlings, that is, the time be- tween arrivals of a parent carrying food for the nestlings. Suppose that the waiting time between arrivals of a parent bird follows the exponential density function f (t) = 0.6e-0,61 for t z 0 and f (t) = 0 for t < 0, where the waiting times are in minutes. (a) What is the mean waiting time? (b) Find the probability that the wait will be between 2 and 5 minutes. (c) Find the probability that the wait will be at least 3 minutes.Suppose X is exponentially distributed with mean 2. Let Y = eX. What is the density function of Y?The life of an automotive component can be modeled by Weibull distributionα =6.2?105 and β =1.3.(i) Calculate the cumulative density function, F (t) Deming believes poor quality was not the fault of workers but as a result of poor management of systems for quality improvement.(ii) Do you agree with the statement? Why do you agree or disagree? (iii) Calculate the reliability function, R (t), and the hazard function, h (t) at the end of the rotating period of 36000 miles.According to a study, the maximum temperature T (in degrees Celsius) of each day during the spring has a distribution with the following function density: 21b. The random variable X has probability density function, f given by E(1+x?), 0< r < 3 f(x) = 0, otherwise Let Y be a random variable such that Y = 3X +1 (i) Find the distribution function of Y i.e. H(y). (the function need not be simplified) (ii) Using this distribution function, determine the 60th percentile of Y. (iii) Hence obtain h(y), the probability density function of Y.Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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