probability考试代写 STAT/MATH 395代写

STAT/MATH 395

Midterm Exam

probability考试代写 We draw a ball from the urn repeatedly with replacement until we see the first green or yellow ball, and then we stop.

Exercise 1.   probability考试代写

An urn contains contains 5 balls: 3 green, 1 yellow and 1 white ones. We draw a ball from the urn repeatedly with replacement until we see the first green or yellow ball, and then we stop. Let X denote the number of draws we needed. Let Y equal to 1 if the last draw is green and 2 if the last draw is yellow.

1. Find the joint distribution of (X, Y ). You can leave your results with combination terms. Hint: consider Y = 1 and Y = 2
2. Find the marginal mass function ofX.
3. Find the marginal mass function of Y.
4. X and Yindependent?

Exercise 2.   probability考试代写

The joint density of X and Y is given by

f (x, y) = 1 (y − x)e^−y, −y < x < y, 0 < y < ∞

1. Find the density function ofX.
2. Find the density function of Y.
3. Are X and Yindependent?

Exercise 3.  probability考试代写

Let X₁, X₂ be independent Exp(λ) random variables. Let U = X₁ + X₂, V = X₁ − X₂.

1. What is the joint pdf of U, V?
2. What is the marginal pdf of V?
3. Are U, Vindependent?

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