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 ).

Expert Solution

Solution: Here the concept of expectation and variance is used.

milcah answered 1 year ago

Derive conditional expectation from joint probability density function.

Using the definition of a young function, prove that the conjugate phi^* of a young function...

Using the definition of a young function, prove that the conjugate phi^* of a young function phi is a young function

. 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

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...

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

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

Let (X, Y) be a random vector with a function of the joint density given by...

Let (X, Y) be a random vector with a function of the joint density given by ˜ fX, Y (x, y) = k (2x + y) I (2,6) (x) I (0.5) (y) a) Determine k so that f X, Y (x, y) is a true probability density function joint quality. b) Determine the marginal probability density functions of X and Y. c) Calculate P (3 <X <4, Y> 2). d) Calculate P (X + Y> 4).

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, Y) be a random vector with a function of the joint density given by...

Let (X, Y) be a random vector with a function of the joint density given by ˜ fX, Y (x, y) = k (2x + y) I (2,6) (x) I (0.5) (y) a) Determine k so that f X, Y (x, y) is a true probability density function joint quality. b) Determine the marginal probability density functions of X and Y. c) Calculate P (3 <X <4, Y> 2). d) Calculate P (X + Y> 4).

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x...

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x + 8 is convex. Is this function strictly convex? 2. Use the definition of convexity to prove that the function f(x)= ax + b is both convex and concave for any a•b ≠ 0.

Question

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

Solutions

Expert Solution

Related Solutions

Derive conditional expectation from joint probability density function.

Using the definition of a young function, prove that the conjugate phi^* of a young function...

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

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

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

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

Let (X, Y) be a random vector with a function of the joint density given by...

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

Let (X, Y) be a random vector with a function of the joint density given by...

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x...