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![Q5:
Two random variables X and Y have a joint density
fx, y(x, y)=[u(x) - u(x-4)]u(y) y³ exp[-(x + 1)²]
Find the marginal densities and distributions of X and Y.
Q6:
4.5-3. Given the joint distribution function
Q8:
Fx, y(x, y) = u(x)u(y)[l-e-a-e-ay+e-ax+]
find:
(a) The conditional density functions fx(x|Y=y) and fy(y)X = x).
(b) Are the random variables X and Y statistically independent?
Q7:
4.5-4. For two independent random variables X and Y show that
PY ≤X) =
x) = [*_ Fr(x) x(x) dx
88
or
P(Y ≤X) = 1-
Fx (1) fy(y) dy
4.5-5. Two random variables X and Y have a joint probability density function.
x²y
0<y<x<2
elsewhere
fx.y(x, y) = 16
(a) Find the marginal density functions of X and Y.
(b) Are X and Y statistically independent?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a72475f-f236-47a3-b928-04aacc1e5dcb%2F790e0cb3-5b4d-4839-a4f2-924cc59bae11%2F2ocqshc_processed.jpeg&w=3840&q=75)
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- X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y22)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?
- X1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is belowa) Find the marginal pmfs of X and Y b) Find the conditional pmf of X given Y = 1LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
- Let X1, ..., Xn be a random sample of size n from a gamma population given by the density f(x; α, β) = ( x α−1e − x β βαΓ(α) if x > 0, α > 0, β > 0 0 if x ≤ 0 with Γ(α) = R ∞ 0 x α−1 e −xdx the well-known gamma distribution. 1. If α > 0 is known for this random sample compute the maximum likelihood estimator βbML estimator of the unknown parameter β > 0. 2. Compute the moment generating function E(e sX) for every s < 1 β . 3. Compute the expectation E(X). 4. Show that the maximum likelihood estimator βbML is a unbiased estimator of β.(Hint: you may use the result of part 2)The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1Let X and Y be random variables with the joint density function f(x,y)=x+y, if x,y element of [0,1], and f(x,y)=0,elsewhere. Find the expected value of the random variable Z = 10X+14Y.
- If two random variables X and Y are independent with marginal pdfs fx(x)= 2x, 0≤x≤1 and fy(y)= 1, 0≤y≤1 Calculate P(Y/X>2)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.)Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
