Derive conditional expectation from joint probability density function.

Expert Solution

orchestra answered 2 years ago

a) Prove, using the joint density function, and the definition of expectation of a function of...

a) Prove, using the joint density function, and the definition of expectation of a function of two continuous random variables (i.e., integration) that E (5X + 7Y ) = 5E (X ) + 7E (Y ). b) (h) Prove, using the joint density function and the definition of expectation of a function of two continuous random variables (i.e., integration) that Var (5X + 7Y ) = 25Var (X ) + 49Var (Y ) + 70Cov (X; Y ).

The random variables X and Y have the following probability density function (pdf). Conditional probability density...

The random variables X and Y have the following probability density function (pdf). Conditional probability density function is f(xly) = ax/y^2 , 0<x<y, 0<y<1 =0 , o/w marginal density finction is f(y) = by^4 ,0<y<1 = 0, o/w At this time, find the constant a and b values to be the probability density function, and then indicate whether the random variables X and Y are independent, and if not, indicate whether they are positive or inverse.

18. Joint Probability Calculations: a) What is a joint probability density function? i. What is an...

18. Joint Probability Calculations: a) What is a joint probability density function? i. What is an example of a discrete joint probability function? ii. What is an example of a continuous joint probability function? b) What are marginal density/mass distribution functions? How can we calculate them from the joint density/mass distribution functions? i. Can you calculate the marginals when the joint space is not a rectangle? (e.g. The space of jpdf[x,y] is 0 < x < y, 0 < y...

. The joint probability density function of X and Y is given by ?(?, ?) =...

. The joint probability density function of X and Y is given by ?(?, ?) = { ??^2? ?? 0 ≤ ? ≤ 2, 0 ≤ ?, ??? ? + ? ≤ 1 0 ??ℎ?????? (a) Determine the value of c. (b) Find the marginal probability density function of X and Y. (c) Compute ???(?, ?). (d) Compute ???(?^2 + ?). (e) Determine if X and Y are independent

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 ??|?? = ??,...

Give an example of a function F which is the joint probability distribution (not density) function...

Give an example of a function F which is the joint probability distribution (not density) function of a pair of random variables X and Y such that (a) X and Y are independent and discrete (b) X and Y are dependent and discrete (c) X and Y are independent and continuous (d) X and Y are dependent and continuous

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...

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 X and Y be a random variables with the joint probability density function fX,Y...

. Let X and Y be a random variables with the joint probability density function fX,Y (x, y) = { 1, 0 < x, y < 1 0, otherwise } . a. Let W = max(X, Y ) Compute the probability density function of W. b. Let U = min(X, Y ) Compute the probability density function of U. c. Compute the probability density function of X + Y ..

Suppose that the joint probability density function of ˜ (X, Y) is given by:´ f X,Y...

Suppose that the joint probability density function of ˜ (X, Y) is given by:´ f X,Y (x,y) = 4x/y3 I(0.1)(x), I (1, ∞)(y). Calculate a) P(1/2 < X < 3/4, 0 < Y ≤ 1/3). b) P(Y > 5). c) P(Y > X).

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