Let X,Y be independent random variables, and Z=X+Y.

Lara Cortez 2022-10-17 Answered
Confusion about sum of random variables conditional probabilities
Let X,Y be independent random variables, and Z = X + Y .. Then I want to calculate P r [ X = x Z = z ] .. My confusion is on evaluating this expression. On the one hand, I have
P r [ X = x Z = z ] = P r [ Z Y = x Z = z ] = P r [ z Y = x ] = P r [ Y = z x ] .
But also, P r [ Z = z X = x ] = P r [ X + Y = z X = x ] = P r [ Y = z x ] . So these two probabilities are equal? But P r [ Z = z X = x ] P r [ X = x ] = P r [ X = x Z = z ] P r [ Z = z ] and in general P r [ X = x ] and P r [ Z = z ] are not equal. I believe it should be P r [ X = x Z = z ] = P r [ Y = z x Z = z ] but I don't think the conditional Z = z can be removed since Y and Z are not independent?
I'm not sure whether the first equation holds either. For example, if I roll a fair six sided die X (numbered 1 to 6) and roll a fair ten sided die Y and take the sum, then P r [ X = 1 Z = 2 ] = 1 since the only possible outcome is ( x , y ) = ( 1 , 1 ) ,, and this is not equal to P r [ Y = ( 2 1 ) ] = 1 / 10.. On the other hand it is equal to P r [ Y = ( 2 1 ) Z = 2 ] = 1.. I think I'm making a mistake in one of these but it's not clear to me in which step.
(The context of this was that X is a random variable with given distribution representing some unknown parameter and Y is a standard normal error. Then you observe z = x + y and want to estimate the X.)
You can still ask an expert for help

Want to know more about Probability?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Ramiro Sosa
Answered 2022-10-18 Author has 13 answers
Step 1
The mistake lies in this step:
P r [ Z Y = x Z = z ] = P r [ z Y = x ]
Note that P ( A | B ) = P ( B | A ) P ( A ) P ( B ) . The actual evaluation is instead
P ( Z Y = x | Z = z ) = P ( Z = z | Z Y = x ) P ( Z Y = x ) P ( Z = z ) = P ( Z = z | X = x ) P ( X = x ) P ( Z = z )
If we are given X = x, Z = z only when Y = z x. Thus, P ( Z = z | X = x ) = P ( Y = z x ). The above formulation should make it clear that this is not true for P ( X = x | Z = z ) . We now have
P ( Z Y = x | Z = z ) = P ( Y = z x ) P ( X = x ) P ( Z = z )
Step 2
In your dice example, this evaluates as
P ( X = 1 | Z = 2 ) = P ( Y = 1 ) P ( X = 1 ) P ( Z = 2 ) = 1 10 1 6 1 60 = 1 as expected.
I shall take another example, to make it clearer. Let us calculate the probability P ( X = 2 | Z = 5 ). We see that Z = 5 is achieved by the following pairs:
( X , Y ) = { ( 1 , 4 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 4 , 1 ) } P ( X = 2 | Z = 5 ) = 1 4
Using the formula, we can calculate the same as
P ( X = 2 | Z = 5 ) = P ( Y = 3 ) P ( X = 2 ) P ( Z = 5 ) = 1 10 1 6 4 60 = 1 4
which is the same as before.
[ P ( Z = 5 ) = 4 60 as there are 60 possible (X,Y) and Z = 5 is only achieved by 4]
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)