Help! Probability Help!Solve Probability Questions: X & Y, X-Y, X+Y

In summary, the conversation discusses the conditional distribution and expectation of X + Y given X − Y for two different scenarios involving independent random variables. The first scenario involves uniform distributions while the second involves exponential distributions. The distributions of X - Y and X + Y given X - Y are also discussed for the second scenario.
  • #1
rk2599
1
0
Help! Probability!

Q1. X and Y are two random variables that are independent and each one is uniform in the interval [01]. The joint density is just 1 in the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 outside. What is the conditional distribution of X + Y given X−Y ? What is the conditional expectation of X + Y given X − Y ?
Q2. If X and Y are independent and each one has an exponential distribution, the joint density is e−(x+y) if x, y ≥ 0 and 0 otherwise. What is the distribution of X − Y ? What is the distribution of X + Y given X − Y ?

Been stuck on this question for hours!
I got 1/2 for the first part of question 1 and 1 for the second part
and I got ∞ for the first part of question 2 and 0 for the second part
 
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  • #2
.Q1. The conditional distribution of X + Y given X−Y is a uniform distribution on the interval [-1, 1]. The conditional expectation of X + Y given X − Y is 0. Q2. The distribution of X - Y is an exponential distribution with parameter 2. The distribution of X + Y given X − Y is also an exponential distribution with parameter 2.
 

FAQ: Help! Probability Help!Solve Probability Questions: X & Y, X-Y, X+Y

What is probability?

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

How do I calculate the probability of X & Y?

The probability of X & Y occurring together is calculated by multiplying the individual probabilities of X and Y. This is known as the multiplication rule of probability.

How do I calculate the probability of X-Y?

The probability of X-Y is calculated by subtracting the probability of Y from the probability of X. This is known as the subtraction rule of probability.

How do I calculate the probability of X+Y?

The probability of X+Y is calculated by adding the probabilities of X and Y and subtracting the probability of X & Y occurring together. This is known as the addition rule of probability.

How can I use probability to solve real-world problems?

Probability can be applied to many real-world scenarios, such as predicting the chances of winning a lottery or the likelihood of a certain event occurring. By understanding probability, you can make more informed decisions and assess risks in various situations.

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