# Finding probability given joint pdf.

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In summary, the problem is asking for the probability of x1 being less than x2, given that x1 is also less than 2x2. The second problem is asking for the probability of x1 being less than x2, and x2 being less than x3, given that x3 is less than 1.
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## Homework Statement

Let X1 X2 X3 be independent and identically distributed random variables with common pdf f(x) = e^-x 0<x<infinity, zero elsewhere.

Evaluate P(X1 < X2|X1 < 2X2) and P(X1 < X2 < X3|X3 < 1).

## The Attempt at a Solution

Would appreciate any hints as to where to start...

I haven't actually studied any statistics so there might be an easier way of doing this but this is how I'd do it...

Having a conditional probability means that you essentially have a 100% probability for the condition, that is

$$\int_{x_1 < 2x_2} dx_1 dx_2 f(x_1, x_2) = 1$$

This gives you the probability density given the condition that x_1 < 2 x_2. You can then use that to find the probability that x_1 < x_2.

clamtrox said:
I haven't actually studied any statistics so there might be an easier way of doing this but this is how I'd do it...

Having a conditional probability means that you essentially have a 100% probability for the condition, that is

$$\int_{x_1 < 2x_2} dx_1 dx_2 f(x_1, x_2) = 1$$

This gives you the probability density given the condition that x_1 < 2 x_2. You can then use that to find the probability that x_1 < x_2.

But I don't think the problem is saying X1 < 2X2 I think the phrasing was X2|X1 < 2X2. Am I thinking about it wrong?

The first problem reads out as "probability that x_1 < x_2, given that you already know that x_1 < 2x_2". The 2nd problem is "probability that x_1 < x_2 < x_3, given that you know that x_3 < 1". Atleast that is what this notation means as far as I know. :-)

## What is a joint PDF and how is it different from a regular PDF?

A joint probability density function (PDF) is a function that describes the probability of multiple random variables taking on specific values simultaneously. It differs from a regular PDF, which describes the probability of a single random variable taking on a specific value.

## How do you find the joint PDF given a set of data?

To find the joint PDF, you first need to determine the probability of each combination of values for the random variables. This can be done by dividing the number of times each combination appears by the total number of data points. Once you have the probabilities, you can create a table or graph to represent the joint PDF.

## What is the formula for finding the probability from a joint PDF?

The formula for finding the probability from a joint PDF is P(A,B) = f(A,B) * ∆A * ∆B, where P(A,B) is the probability of the random variables A and B taking on specific values, f(A,B) is the joint PDF, and ∆A and ∆B are small intervals around the values of A and B.

## What is the difference between a discrete and continuous joint PDF?

A discrete joint PDF is used when the random variables can only take on a finite number of values, while a continuous joint PDF can be used when the random variables can take on any value within a given range. This means that a discrete joint PDF would use a table or graph to represent the probabilities, while a continuous joint PDF would use a function or equation.

## How is the concept of independence related to joint PDF?

If two random variables are independent, then their joint PDF can be expressed as the product of their individual PDFs. This means that the probability of one variable taking on a specific value is not affected by the value of the other variable. In contrast, if two variables are not independent, their joint PDF cannot be expressed as the product of their individual PDFs.

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