Find Density Functions of X, Y, Z Variates

In summary, the random variable X assumes the values 1,2,3 and 4 with equal probability. The density functions for the variates Y=1-2X and Z=X/(X+1) are 1/4 for each value of Y and Z. The exam went well except for one confusing problem. It's important to study the pdf of a random variable of a decreasing function, not just an increasing one. There was no extra credit and only five out of six problems were graded.
  • #1
iHeartof12
24
0
The random variable X assumes the values 1,2,3 and 4 with equal probability. Find the density functions of the following variates:

Attempted solutions:

X 1 2 3 4
Pr(X) 1/4 1/4 1/4 1/4

a) Y=1-2X

Y -1 -3 -5 -7
Pr(Y) 1/4 1/4 1/4 1/4

b) Z= X/(X+1)

Z 1/2 2/3 3/4 4/5
Pr(Z) 1/4 1/4 1/4 1/4

Are my solutions for parts a and b correct?
 
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  • #2
Yes, this is correct.
 
  • #3
Thank you. How'd your exam go today? I'm ready to get my exam over tomorrow I have a few more next week that I'm also studying for. Ahh the life of a college student. lol.
 
  • #4
It went fine except for one problem that made absolutely no sense to me. Don't forget to study the pdf of a random variable of a decreasing function, not just an increasing one. There was no extra credit :( But I guess that's what you get when you can do six and he'll grade your best five.
 

FAQ: Find Density Functions of X, Y, Z Variates

1. What is a density function?

A density function is a mathematical representation of the probability distribution of a random variable. It describes the relative likelihood of a variable taking on a particular value or range of values.

2. How is the density function of a random variable calculated?

The density function of a random variable is calculated by taking the derivative of the cumulative distribution function (CDF) for continuous variables or by summing the probabilities for each possible value for discrete variables.

3. What is the difference between a probability density function (PDF) and a cumulative distribution function (CDF)?

A PDF describes the probability of a variable taking on a specific value, while a CDF describes the probability of a variable being less than or equal to a specific value. In other words, a PDF gives the exact probability at a specific point, while a CDF gives the overall probability up to that point.

4. How is the density function of multiple variables (X, Y, Z) determined?

The density function of multiple variables is determined by finding the joint probability distribution function for all of the variables. This involves finding the probability of all variables taking on specific values simultaneously.

5. Why is it important to find the density function of random variables?

Finding the density function of random variables is important because it allows us to make predictions about the behavior of the variables and calculate probabilities for different outcomes. It is also a fundamental concept in many statistical and mathematical models used in various fields of science.

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