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- The number of minutes that train is early or late can be modeled by a random variable whose density is given by: g(t) = {(1/972)(81 − t^2), −9 ≤ t ≤ 9, 0 elsewhere, where negative values indicate the train arriving early and positive values indicate the train arriving late. a. Find the probability that one of the train trips will arrive more than 5 minutes early. b. Find the probability that one of the train trips will arrive between 1 and 8 minutes late. c. Without integration, give the expected number of minutes early/late. Examination of the graph and recollection of properties of integrals will allow this.A given car company set a standard that for each tire should be filled to a pressure of 40 psi. Given X and Y are random variables and X denotes the actual pressure on the front tires and Y denotes the actual pressure on the back tires. Their joint density function is given by: (in the picture) a. Find the value of the constant Q.b. Find P(30 ≤ X ≤ 40 , 40 ≤ Y ≤ 50)c. Find the probability that both of the tires have insufficient amount of pressure.the popularity density function f(x) for a uniform random variable X defined over the interval [2,10] is?
- An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the chemical is first determinedto be present. At very low concentrations, however, the chemical may fail tobe detected even if it is present. This phenomenon is modeled by assuming thatif the concentration is x, the chemical is detected with probability R(x). Let Ydenote the concentration of a chemical in a soil in which it has been determinedto be present. Show that the density function of Y isg(y) = R(y) f (y)/An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the…Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.According to the Maxwell–Boltzmann law of theoret-ical physics, the probability density of V, the velocity of a gas molecule, isf(v) =⎧⎨⎩kv2e−βv2for v > 00 elsewhere where β depends on its mass and the absolute tem-perature and k is an appropriate constant. Show that the kinetic energy E = 1 2mV2, where m the massof the molecule is a random variable having a gammadistribution.
- The probability density function of the random variable X is as in the picture with λ> 0. Find the maximum likelihood estimator (λ^) of the parameter λ.If the total nunber of hours spent by a student to study his/her module over one day is a continuous random variable X that has the density function *refer from attached* Find the probability that in one day, the student is studying a) less than 1 hr b) between 2 and 3hrs c) more than 3hrsIf a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function Find the variance of X.
- A given car company set a standard that for each tire should be filled to a pressure of 40 psi. Given X and Y are random variables and X denotes the actual pressure on the front tires and Y denotes the actual pressure on the back tires. Their joint density function is given by: a. Find the value of the constant Q.b. Find P(30 ≤ X ≤ 40 , 40 ≤ Y ≤ 50)c. Find the probability that both of the tires have insufficient amount of pressure.Consider the random variable model with density function given byf(x) ={2kx⁄45 , 0 ≤ x ≤ 15 2k(30 − x)/45 , 15 ≤ x ≤ 300, otherwise Compute, Comment on the output of these results below, what will bethe conclusion of your output.(i) k(ii) expected value (iii) standard deviationA particular pumping engine will only function properly if an essential component functions properly. The time to failure of the component ( in thousands of hours) is a random variable X with probability density f(x) = 0.02xe-0.01x^2 for x > 0. What is the proportion of pumping engines that will not fail before 10,000 hours of use? What is the probability that the engine will survive for another 5000 hours, given that it has functioned properly during the past 5000 hours?
