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- Suppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain plant has a standard beta distribution with ? = 4 and ? = 2. (a) Compute E(X) and V(X). (Round your answers to four decimal places.) E(X) = V(X) = (b) Compute P(X ≤ 0.3). (Round your answer to four decimal places.)(c) Compute P(0.3 ≤ X ≤ 0.7). (Round your answer to four decimal places.)For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)The Rayleigh density function is as shown in the image (0, for any other value)A random sample of size n = 4 is drawn from a population with a Rayleigh distribution and the following results are obtained:y1 = 2.39 ; y2 = 1.59 ; y3 = 1.52 ; y4 = 1.35.What is the maximum likelihood estimator for this sample?
- Find the pdf of e^x in terms of the pdf of X, base the answer to the case where X is uniformly distributed between 0 and 1Consider random variables Xand Y with following joint pdf given as f(x,y) ={x+y 0≤x≤1, 0≤y≤1, 0 elsewhere. Compute correlation coefficient, ρXYLet X1, X2, ..., Xn be a random sample from the distribution with the pdf f(x; theta) = theta*x^(theta-1), 0<x<1, 0<theta<infinity, zero elsewhere. Find the mle theta-cap of Theta.
- The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function. What is P(X ≤ 4)? What is P(4.5 < X)?Suppose that the continuous random variable X has CDF Fx(X) = {(x-2)/x , x>2 and 0, x=<2} a. Determine, and sketch, the pdf (probability density function) of X. b. Find the mean and variance of X c. Determine the pdf of the random variable Y=X^2Suppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x) = 1/θ if 0 ≤ x ≤ θ, where θ > 0 is an unknown parameter. (a) Find E(X1) (b) Find the MME for θ. (c) Compute the variance of your MME from part (a).
- Consider X and Y are joint distributed with PDFf(x,y)=x+y, 0≤x≤1, 0≤y≤1. (a) Find the probability that X > 0.5. (b)FindP(X> Y 1/2).(c) Are X and Y independent?Which of the following correctly compares the tt-distribution and zz-distribution? A. For small sample sizes, the density curve of the t-distribution is not symmetric, but the density curve of the z-distribution is symmetric. B. For large sample sizes, the density curve of the t-distribution is not symmetric, but the density curve of the z-distribution is symmetric. C. The curves of both distributions are symmetric, but the height of the density curve of the t-distribution is taller than the height of the density curve of the z-distribution. D. The area under the density curve of the t-distribution is greater than the area under the density curve of the z-distribution, especially for small sample sizes. E. The density curve of the t-distribution is more spread out than the density curve of the z-distribution, especially for small sample sizes.Given the function y = x2 considered as a probability density distribution defined at 0 ⩽ x ⩽ 3, find the median and the first and third quartiles from the integral function