Did you meanwhitejac said:Homework Statement
Given a Uniform Distribution (0,1) and Z = X/Y
Find F(z) and f(z)
Homework Equations
The Attempt at a Solution
So I'm just trying to make sure i have the range correct on this one... I'm honestly lost from beginning to end with it.
R(z) = {0,∞} because as y is very small, Z becomes very big.
After that, I'm not quite sure though because that would mean that P(z) would be 1 for z > ∞, and that doesn't make much sense...
whitejac said:Homework Statement
Given a Uniform Distribution (0,1) and Z = X/Y
Find F(z) and f(z)
Homework Equations
The Attempt at a Solution
So I'm just trying to make sure i have the range correct on this one... I'm honestly lost from beginning to end with it.
R(z) = {0,∞} because as y is very small, Z becomes very big.
After that, I'm not quite sure though because that would mean that P(z) would be 1 for z > ∞, and that doesn't make much sense...
The Probability Density Function (PDF) of a given function Z = X/Y is a mathematical expression that describes the probability of a continuous random variable Z taking on a certain value, based on the values of the variables X and Y. It is denoted as f(z) and is calculated by taking the derivative of the Cumulative Distribution Function (CDF) of Z.
The Cumulative Distribution Function (CDF) of a given function Z = X/Y is a mathematical expression that describes the probability of a random variable Z being less than or equal to a certain value. It is denoted as F(z) and is calculated by taking the integral of the Probability Density Function (PDF) of Z.
The main difference between the PDF and CDF of a given function Z = X/Y is that the PDF gives the probability of a specific value of Z occurring, while the CDF gives the probability of Z being less than or equal to a certain value. In other words, the PDF describes the probability distribution of Z, while the CDF gives the cumulative probability distribution of Z.
Yes, for example, let's say we have the function Z = X/Y, where X and Y are independent random variables with a normal distribution. In this case, the PDF of Z would also follow a normal distribution, and the CDF of Z can be calculated by taking the integral of the PDF. The specific values of X and Y would determine the mean and standard deviation of the normal distribution for Z.
Finding the PDF and CDF of a given function Z = X/Y is important because it allows us to understand and analyze the probability distribution of Z. This information is useful in a variety of fields, such as finance, engineering, and statistics, to make informed decisions and predictions based on the values of X and Y. It also helps us to determine the likelihood of a specific event occurring and to calculate important metrics such as the mean, variance, and standard deviation of Z.