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- Suppose that Y1, . . . , Yn is a random sample from a population whose density function isSuppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with density curve f(x) = (1/3)e(-x/3) for x > 0 and f(x) = 0 otherwise. What is the probability that response time exceeds 5 seconds? What is the probability that response time exceeds 10 seconds?Suppose that X ~ Unif[0, 1] and let Y = e^X. (a) Find the cumulative distribution function of Y .(b) Find the probability density function of Y .
- Suppose that X and Y have a joint probability density function f(x,y)= 1, if0<y<1,y<x<2y; 0, otherwise. (a) Compute P(X + Y less than or equal 1). (b) Find the marginal probability density functions for X and Y , respectively. (c) Are X and Y independent?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.)Suppose X and Y are independent and identically distributed (i.i.d.) randomvariables, each with the uniform distribution on [0, 1]. What is the cumulative distributionfunction and the density function of XY ?
- With the known values of a = 200,000 and b = 230,000, the first equation for the probability density function for the sales price of a home is found as follows. f(x) = 1/b – a, a ≤ x ≤ b = 1/230,000 − 200000, 200,000 ≤ x ≤ 230,000 = 1/30000, 200,000 ≤ x ≤ 230,000 Everywhere else, the probability density function will just be 0. Therefore, the full probability density function for the sales price of a home follows. f(x) = _______________ 200,000 ≤ x ≤ 230,000 elsewhereSuppose that the unknown X is ≥ 2 and has the probability density function fX(x)=Ce^−x,x ≥ 2.What is the numerical value of C?Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?
- let x denotes the percentage of time out of 40 hour workweek that a call center agent is serving a client by answering phone calls, suppose that x has probability density function defin by f(x)=3x^2 for 0< x < 1. find the mean and variance of xSuppose that the joint probability density function of X and Y is fX,Y(x,y) = 10.125(x2 – y2) e−3x , for 0<x<∞ and -x<y<x 0, otherwise Give your answers to the below questions in two decimal places where appropriate. (a) The marginal probability density function of X is given by: fX(x) = A xB e-3x , for 0<x<∞ 0, otherwise Find the value of A. (b) Find the value of B. (c) The conditional probability density function of Y, given that X=x for some x>0, takes the following form: fY|X=x(y) = C (x2-y2) xD e-Ex, for -x<y<x 0, otherwise. Find the value of C (d)Find the value of D. (e)Find the value of E.Suppose that the joint density function of the random variables X and Y is f(x,y)=k(1+2y), if 1<x<13 and 0<y<1, and f(x,y)=0, otherwise. Show that the marginal distribution of X is g(x)=c, if 1<x<13, and g(x)=0 otherwise. Enter the value of c. Hint: Of course, first, you need to find the value of k. Round your answer to a number with two decimal digits after the decimal point. For example if your answer is 1/40, which is equal to 0.025, then you should enter 0.03. (Do NOT use decimal comma; 0,03 would be wrong.)
