代写ECE2191 Probability Models in Engineering Tutorial 8代写留学生Matlab语言
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Tutorial 8: Independent RVs and Jointly Gaussian RVs
Second Semester 2021
1. Suppose X and Y are random variables with the following joint PMF. Are X and Y independent?
2. Suppose X and Y are random variables with
Let c := P (X = 1, Y = 1).
(a) Determine the joint distribution of X and Y , R X,Y, Cov(X, Y) and PXY .
(b) For what value(s) of c are X and Y independent? For what value(s) of c are X and Y 100% correlated?
3. In this problem, we want to explore the relationship between correlation and dependence of random variables.
(a) Prove that if two random variables are independent, then they are uncorrelated.
(b) Consider random variables X and Y with the following joint PMF. Show that X and Y are uncorrelated but dependent.
(c) What is your conclusion from part (a) and (b)?
4. Determine whether X and Y are independent:
(a)
(b)
5. Let X and Y be jointly continuous random variables with joint PDF
(a) Are X and Y independent?
(b) Find E[Y|X > 2].
(c) Find P (X > Y).
6. Let X and Y be two uncorrelated Gaussian random variables. Show that they are inde- pendent as well.
7. Let N be the total number of people that go to get tested for COVID-19 at a testing center during a day, where we know that N follows a Poisson distribution with parameter λ . Each test is positive with probability p and is negative with probability 1 - p independent of others.
(a) What is the probability that the testing center receives exactly x positive and y negative tests during a day?
(b) Let X and Y be two random variables denoting the number of positive and negative cases during a day. Are they independent?
8. Derive the CDF and PDF of Z = X +Y, where X and Y are two identical and independent uniform random variables in the [0, 1] range.
9. Let X and Y be a pair of jointly Gaussian random variables. Then, given that X = x, show that Y is a Gaussian random variable with
10. Let X and Y be jointly normal random variables with parameters µX = 1, = 1, µY =
0, = 4, and
(a) Find P (2X + Y ≤ 3).
(b) Find Cov(X + Y, 2X - Y).
(c) Find P (Y > 2|X = 2).
Hint: You can use the fact that the linear combination of Gaussian random variables is Gaussian.
11. [Optional] Consider a disk with radius l. A single point from the surface of this disk is selected randomly and uniformly. Therefore, the probability distribution of the chosen point both in the polar coordinates and the Cartesian coordinates is given by
.
(a) What is the marginal PDF of X?
(b) What is the expected value of R?
12. [Optional] In this problem, we want to explore the properties of moment generation functions. (Hint: A moment generating function definition and equation can be found towards the end of page 62 from the lecture notes.) Show that:
(a) If Y = aX + b, then MY(s) = esbMX(as).
(b) If X and Y are independent, then MX+Y(s) = MX(s)MY(s).
(c) Let X and Y be independent random variables. Let Z be equal to X with probability p, and equal to Y with probability 1 - p. Then, MZ(s) = pMX (s) + (1 - p)MY(s).