Finding the PDF of X Given an Expression

In summary: I think you may have accidentally used lowercase ##x_3## in the first argument of the inner Pr() function. If you fix that, your solution is correct.In summary, the conversation discusses finding the probability density function (pdf) of an expression involving three independent and identically distributed exponential random variables. The correct method involves using the cumulative distribution function (cdf) and factoring the joint pdf into the individual pdfs of the random variables. However, one of the lower integration limits needs to be corrected to include ##x_2=0##. It is also recommended to use uppercase for random variables to avoid confusion with quantiles.
  • #1
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Hi.

I have the following expression

[tex]X=\frac{G\frac{x_1}{x_2}\gamma_Q}{\frac{1}{G}x_3 \gamma_P+1}[/tex]

where ##x_i## is an exponential random variable with mean 1. All other parameters are nonzero positive constants. Basically I want to find the probability density function (pdf) of ##X##. So I started with cumulative distribution function (cdf) as

[tex]F_X(x)=\text{Pr}\left[\frac{G\frac{x_1}{x_2}\gamma_Q}{\frac{1}{G}x_3 \gamma_P+1}\leq x\right][/tex]

which I evaluated it as

[tex]F_X(x)=\int_{x_3=0}^{\infty}\int_{x_3=0}^{\infty}\text{Pr}\left[x_1\leq\frac{x_2\,x}{G \gamma_Q}(\frac{1}{G}x_3 \gamma_P+1)\right]f_{X_2}(x_2)f_{X_3}(x_3)\,dx_2\,dx_3[/tex]

where ##f_{X_i}(x_i)## is the pdf of the random variable ##x_i##. Is what I did correct?

Thanks
 
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  • #2
It is only correct if the three random variables are independent. If they are, it is valid to factorise the joint pdf ##f_{X_1,X_2,X_3}(x_1,x_2,x_3)## into ##f_{X_1}(x_1)f_{X_2}(x_2)f_{X_3}(x_3)## as you have done. If they are not independent, that factorisation is incorrect.

Also, one of your two lower limits for integration needs to be ##x_2=0##.
 
  • #3
andrewkirk said:
It is only correct if the three random variables are independent. If they are, it is valid to factorise the joint pdf ##f_{X_1,X_2,X_3}(x_1,x_2,x_3)## into ##f_{X_1}(x_1)f_{X_2}(x_2)f_{X_3}(x_3)## as you have done. If they are not independent, that factorisation is incorrect.

Also, one of your two lower limits for integration needs to be ##x_2=0##.

Yes, they are independent and identically distributed random variables. Right, the first integral is over all values of ##x_2##, it's a typo. Thanks
 
  • #4
S_David said:
Yes, they are independent and identically distributed random variables. Right, the first integral is over all values of ##x_2##, it's a typo. Thanks
Given that additional info, and the discussed correction to the lower integration limit, what you've written is correct, but it is better practice to use upper case for random variables, to distinguish them from quantiles of those variables, ie:

$$F_X(x)=\int_{x_3=0}^{\infty}\int_{x_2=0}^{\infty}\text{Pr}\left[X_1\leq\frac{x_2\,x}{G \gamma_Q}(\frac{x_3 \gamma_P}{G}+1)\right]f_{X_2}(x_2)f_{X_3}(x_3)\,dx_2\,dx_3$$
 
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1. What is a PDF and why is it important in scientific research?

A PDF (Probability Density Function) is a mathematical function that describes the probability of a random variable taking on a certain value. It is important in scientific research because it allows scientists to understand the likelihood of certain events occurring and make predictions based on data.

2. How do you find the PDF of X given an expression?

To find the PDF of X given an expression, you can use mathematical techniques such as integration and differentiation. First, you need to determine the bounds of the random variable X and then use the expression to calculate the probability density at each point within those bounds.

3. Can you provide an example of finding the PDF of X given an expression?

Sure, let's say we have a random variable X representing the heights of people in a population. We have collected data and determined that the average height is 5 feet and the standard deviation is 2 inches. We can use the expression for a normal distribution to find the PDF of X given an expression: f(x) = (1/(2√π)) * e^(-(x-5)^2/8).

4. How is the PDF of X related to other statistical measures such as mean and variance?

The PDF of X is related to other statistical measures such as mean and variance through the mathematical formulas used to calculate them. For example, the mean of a random variable X can be calculated by integrating its PDF over its entire range of values. The variance of X can be calculated using the mean and the PDF as well.

5. Are there any limitations to finding the PDF of X given an expression?

Yes, there are limitations to finding the PDF of X given an expression. It can be difficult to find an exact expression for the PDF in complex situations, and sometimes numerical methods must be used instead. Additionally, the expression used to calculate the PDF may not accurately represent the true distribution of the data, leading to errors in predictions and analysis.

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