Break a Stick Example: Random Variables

In summary, the conversation involves a request for feedback on a question related to random variables. The speaker suggests approaching the problem by calculating the cumulative probability function and then the PDF, but notes that in simpler cases, one can skip these steps. They also advise drawing a diagram of the integration region on an x-u plane to better understand the problem.
  • #1
ashah99
60
2
Homework Statement
Provided below in the comments
Relevant Equations
f(x,y) = f(y|x)f(x)
Law of iterated expectation: E[ Y ] = E( E(Y|X) )
Hello, I would like to confirm my answers to the following random variables question. Would anyone be willing to provide feedback and see if I'm on the right track? Thank you in advance.

1663188804733.png

My attempt:

1663189296923.png
 
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  • #2
You have not explained your choice of integration limits in (a), and I think the set-up of the integral is wrong.
One usually approaches problems like this by calculating a formula for the cumulative probability function ##F_Y(y) = Prob(Y\leq y)##, and then calculating the PDF as ##f_Y(y) = \frac d{dy} f_Y(y)##. Only in very simple cases can you skip these steps and go directly to the PDF.
The formula for ##F_Y(y)## will be a double integral over ##x\in(0,1)## and ##u\in[0,\min(x,y)]##. To start, draw a diagram of the integration region on an x-u plane. You will see that the region has a trapezium shape, so you will need to break the inner integral into two parts.
 

FAQ: Break a Stick Example: Random Variables

1. What is a random variable?

A random variable is a numerical quantity that takes on different values based on the outcome of a random event. It is often denoted by the letter X and can be discrete or continuous.

2. What is the "break a stick" example of a random variable?

The "break a stick" example is a simple experiment where a stick is broken into two pieces at a random point. The lengths of the two pieces can be represented by a random variable, with the possible values being any number between 0 and the length of the original stick.

3. How is a probability distribution used in the "break a stick" example?

A probability distribution is used to describe the likelihood of different values occurring for the random variable in the "break a stick" example. This can be visualized with a graph or table, where the x-axis represents the possible values and the y-axis represents the probability of each value occurring.

4. What is the expected value in the "break a stick" example?

The expected value in the "break a stick" example is the average value that we would expect to get if we repeated the experiment many times. It is calculated by multiplying each possible value by its corresponding probability and summing all of these products.

5. How can the "break a stick" example be applied in real life?

The "break a stick" example can be applied in real life situations where there is uncertainty or randomness involved. For example, it can be used to model the lengths of broken bones in a medical study or the lengths of cracks in a material under stress.

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