Example 9-53. (a) Find the p.d.f. of X) in a random sample of size n from the exponential distribution: f(x) = a eax , a>0, x 2 0. (6) Show that X and W, = X9- X¢, r
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- Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxy
- 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter α and rate parameter β. Given Y = y, the number of accidents that the individual suffers in years 1, 2, . . . , n are independent random variables X1, X2, . . . Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X1, . . . , Xn can be described via P(a ≤ Y ≤ b, X1 = k1, X2 = k2 . . . Xn = kn) = Z b a f(y, k1, k2, . . . kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X1, X2, . . . Xn are independent, each having the Poisson distribution with parameter y. (b) Find the conditional distribution of Y given that X1 = k1, X2 = k2, . . . , kn. (c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to…Suppose the lifespan (in months) of a smartphone battery can be modeled as a continuous random variable with CDF F(x) = 1 − e-x/3 x ≥ 0 What is the probability that the battery lasts between 12 to 15 months?Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of X1/(X1+X2+X3+X4) and (X1+X2)/(X1+X2+X3+X4) and (X1+X2+X3)/(X1+X2+X3+X4) using Jacobian method?
- Data collected over a number of years show that whena broker called a random sample of eight of her clients,she got a busy signal 6.5, 10.6, 8.1, 4.1, 9.3, 11.5, 7.3, and5.7 percent of the time. Assuming that these figures canbe looked upon as a random sample from a continuous uniform population, use the estimators obtained in Exer-cise 57 to estimate the parameters α and β.Let Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?At a toll-booth on-ramp there is a stochastic arrival distribution. Vehicles are counted in 20-second intervals, and vehicle counts are taken in 120 of these time intervals. Based on data collected, no vehicles arrived in 18 of the 120 count intervals. What is the number of the 120 intervals that 3 cars arrived?
- Suppose that X1,X2,...,X200 is a set of independent and identically distributed Gamma random variables with parameters α = 4,λ = 3. Describe the normal distribution that would correspond to the sum of those 200 random variables if the Central Limit Theorem holds.Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter (lambda) =3.1 Find the probability that 3 or more accidents occur today Find the probability that no accidents occur todayA random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance ofYY if only 2.5% of its elements is a subset of X.