Finding probability given joint pdf.

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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...
 

Answers and Replies

  • #2
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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

[tex] \int_{x_1 < 2x_2} dx_1 dx_2 f(x_1, x_2) = 1 [/tex]

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.
 
  • #3
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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

[tex] \int_{x_1 < 2x_2} dx_1 dx_2 f(x_1, x_2) = 1 [/tex]

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?
 
  • #4
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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. :-)
 

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