The continuous random variables ? and ? have the joint p.d.f. ??,? given by ?(?, ?)...

The continuous random variables ? and ? have the joint p.d.f. ??,? given by

?(?, ?) = { ?^ (−?−?) , ? > 0, ? > 0

0 otherwise

a) Find the conditional p.d.f. ??|?(?|?) of ? given ? = ?. [1]

b) Find ?(?|? = ?) and ???(?|? = ?). [1]

c) Find ?(?) and ???(?). [1]

d) Find ???(?, ?). [1]

Expert Solution

milcah answered 11 months ago

The joint pdf of a two continuous random variables is given as follows: ??,? (?, ?)...

The joint pdf of a two continuous random variables is given as follows: ??,? (?, ?) = { ??? 0 < ? < 2, 0 < ? < 1 0 ??ℎ?????? 1) Find c. 2) Find the marginal PDFs of ? and ?. Make sure to write the ranges. Are these random variables independent? 3) Find ?(0 < ? < 1|0 < ? < 1) 4) What is ??|? (?|?). Make sure to write the range of ?.

6. Two continuous random variables X and Y have the joint density that is equal to...

6. Two continuous random variables X and Y have the joint density that is equal to c(x + y) 2 inside the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and vanishes everywhere else. a) Find the value of c b) Find the marginal density fX(x) and use it to determine E(X) and V(X) c) Find the marginal density fY(y) and use it to determine E(Y) and V(Y) d) Find the correlation ρ(X,Y) e) Are X and...

The joint probability density function for two continuous random variables ?? and ?? is ??(??, ??)...

The joint probability density function for two continuous random variables ?? and ?? is ??(??, ??) = ??(3??2 − ??), 0 < ?? < 1, 0 < ?? < 1. Answer the following: (a) Find the value of ?? such that ??(??, ??) is a valid probability density function. (b) Find the marginal probability density function for ??, ??(??). (c) Find the marginal probability density function for ??, ??(??). (d) Find the conditional probability density function for ??|?? = ??,...

The random variables ? and ? have the following joint pdf. ??,? (?, ?) = ??...

The random variables ? and ? have the following joint pdf. ??,? (?, ?) = ?? -8x^2-18y^2 a) Find the mean and variance of ? and ? and the value of ?. b) Determine if ? and ? are independent. c) Determine the distribution of ? and ?.

Let X and Y be two continuous random variables with the joint probability density function of...

Let X and Y be two continuous random variables with the joint probability density function of for 0 < x < 2, 0 < y < 2, x + y < 1,where c is a constant. (In all the following answers, you do NOT need to find what the value of c is; just treat it as a number.) (a) Write out the marginal distribution of Y. (b) P(Y < 1/3) = ? (c) P(X < 1.5, Y < 0.5)=...

Let ?1 , ?2 , ... , ?? be independent, identically distributed random variables with p.d.f....

Let ?1 , ?2 , ... , ?? be independent, identically distributed random variables with p.d.f. ?(?) = ???−1, 0 ≤ ? ≤ 1 . c) Show that the maximum likelihood estimator for ? is biased, and find a function of the mle that is unbiased. (Hint: Show that the random variable −ln (??) is exponential, the sum of exponentials is Gamma, and the mean of 1/X for a gamma with parameters ? and ? is 1⁄(?(? − 1)).) d)...

If the joint probability density function of the random variables X and Y is given by...

If the joint probability density function of the random variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x < 2, 0 < y < 1, 0 elsewhere (a) Find the conditional density of Y given X = x, and use it to evaluate P (X + Y/2 ≥ 1 | X = 1/2) (b) Find the conditional mean and the conditional variance of Y given X = 1/2 (c) Find the variance...

Which of these variables are discrete variables, and which are continuous random variables?

Which of these variables are discrete variables, and which are continuous random variables?The number of new accounts established by a salesperson in a year.The time between customer arrivals to a bank automatic teller machine (ATM).The number of customers in Big Nick’s barber shop.The amount of fuel in your car’s gas tank last week.The outside temperature today.

please show all steps Q.4 Let X and Y be continuous random variables with the joint...

please show all steps Q.4 Let X and Y be continuous random variables with the joint pdf: f(x, y) = { k(x + y), if (x, y) ∈ 0 ≤ y ≤ x ≤ 1 ; 0 otherwise Answer the question with the equation below please f(x, y) = 2(x + y), for 0 ≤ y ≤ x ≤ 1 and 0 otherwise. (a) Find E[X + Y ] and E[X − Y ] (b) Find E[XY ] (c) Find...

Let X and Y be continuous random variables with joint pdf f(x, y) = kxy^2 0...

Let X and Y be continuous random variables with joint pdf f(x, y) = kxy^2 0 < x, 0 < y, x + y < 2 and 0 otherwise 1) Find P[X ≥ 1|Y ≤ 1.5] 2) Find P[X ≥ 0.5|Y ≤ 1]

Question