How to find this particular probability?

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In summary, the problem is to find the probability P(X1+X2<X3) when three statistically i.i.d continuous random variables X1, X2, X3 each are uniformly distributed in the range [0,1]. The solution involves setting up a triple integral with appropriate limits of integration based on the given conditions. The limits of integration for the outer integral are 0 and 1. The limits for the second integral depend on the value of x3, while the inmost integral has limits of 0 and x3-x2. These limits are determined by the intersection of the conditions x1+x2<x3 and each xi being between 0 and 1. The solution can also be obtained using a software such
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dexterdev
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Homework Statement



Suppose there are three statistically i.i.d continuous random variables X1, X2, X3 each are uniformly distributed in the range [0,1]. How to find the probability P(X1+X2<X3)?

Homework Equations


The below given equations are the steps to the solution. But I can't figure out how the limits of integral comes this way.

[itex] \int_0^1 \int_0^{x_3}\int_0^{x_3-x_2} \,dx_1\,dx_2\,dx_3 =\int_0^1 \int_0^{x_3} (x_3-x_2) dx_2\,dx_3 = \int_0^1 x_3^2 - \frac{x_3^2}{2}\,dx_3 = \frac16 = 0.1\overline 6[/itex]


The Attempt at a Solution



I tried this using a software called MATLAB by generating three pseudo random variables (1000 samples) and finding X1+X2−X3 and plotting its CDF through a MATLAB tool called dfittool. I got the answer around 0.169. But how do I do this theoretically? Especially how to figure out the limits in those integrals?
 
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  • #2
Your questions make no sense: you have already obtained the answer theoretically, and you have already written the limits of integration.
 
  • #3
@Ray Vickson : Yes I have got those limits from someone else, but never told how they come?
 
  • #4
They come from two different concerns, that x1+x2<x3 and that each xi must be between 0 and 1. Those integration limits represent the intersection of those concerns.
 
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  • #5
You are told that the three variables all lie in [0, 1]. The limits on the outer integral, with respect to [itex]x_3[/itex] must be constants so must be 0 and 1. The next inner integral can have limits depending on [itex]x_3[/itex]. Since we have [itex]x_1+ x_2< x_3[/itex] and [itex]x_1[/itex] can be 0, [itex]x_2[/itex] can go from 0 to [itex]x_3[/itex]. Finally, [itex]x_1+ x_2< x_3[/itex] means that [itex]x_1< x_3- x_2[/itex] so the inmost integral has limits of 0 to [itex]x_3- x_2[/itex].
 
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FAQ: How to find this particular probability?

What is probability and why is it important in science?

Probability is a measure of the likelihood of an event occurring. It is important in science because it allows us to make predictions and draw conclusions based on data and evidence.

How do you calculate the probability of an event?

To calculate the probability of an event, you divide the number of favorable outcomes by the total number of possible outcomes. This is known as the classical probability formula.

What is conditional probability and how is it different from regular probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It is different from regular probability because it takes into account additional information or conditions.

How do you use a probability distribution to find a particular probability?

A probability distribution is a function that assigns probabilities to all possible outcomes of an event. To find a particular probability, you would locate the specific outcome on the distribution and determine its corresponding probability.

Can you give an example of how to find a particular probability using Bayes' theorem?

Bayes' theorem is used to calculate conditional probabilities. For example, if you want to find the probability of a person having a certain disease given that they have tested positive for it, you would use Bayes' theorem to calculate this conditional probability using the individual's test results and the prevalence of the disease in the population.

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