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I've posted this on other forums and have had little to no help. I thought this forum looked very good so hoping someone can help.

Sorry for the non specific title of the thread, I don't know what the following problem should be called! I don't think it is too hard but I stopped studying probability after my first year at uni and do not have much intuition in this field of mathematics. Anyway, here goes...

If X_{1}and X_{2}are random variables that can take any values in the finite set {0,2,4,...,2n} all with equal chance then how do I work out the probability, given an arbitrary positive real a that:

X_{1}<= a <= X_{1}+X_{2}

If we call this probability P(a), how do we find a such that

P(a) => P(b) for arbitrary positive real b

Acting on advice I have calculated the pmf for X_{1}and Y= X_{1}+ X_{2}

X_{1}: S ->Ris defined by th identity function X_{1}(s) = s.

Hence, f_{1}(x) = P(X_{1}= x) = P({s € S: X_{1}(s) = x}) = P({s€S: s = x) = 0 if x does not belong to {0,2,...,2n}; 1/(n+1) if x € S

Since X_{2}: S ->Ris defined by X_{2}(s) = s we have f_{1}(x) = f_{2}(x).

Let Y = X_{1}+ X_{2}. Y: SxS ->Ris defined by Y(s,s') = X_{1}(s) + X_{2}(s') = s + s'.

Hence f_{Y}(x) = P(Y = x) = P({(s,s') € SxS : Y(s,s') = x}) = P({(s,s') € SxS: s + s' = x) = 0 if x does not belong to {0,2,...,4n}; (x+2)/2(n+1)^{2}if x € {0,2,...,2n}; (2(2n+1)-x)/2(n+1)^{2}if x € {2n, 2n+2,..., 4n}

However, after this no further advice followed.

In the process of trying to solve it my self and guessing what the further advice may have been I have calculated explicitly the cdf for X_{1}and Y = X_{1}+ X_{2}which I shall denote F_{X1}(x) and F_{Y}(x) respectively.

Let E be the event X_{1}<= x, E' the event x <= Y.

Now if E and E' were independant by definiton P(E and E') = P(E)P(E') = F_{X1}(x)(1-F_{Y}(x)+f_{Y}(x)) and hence the problem would be solved. However, E' is conditional on E. I would go on to solve this problem by working out f_{Y/E}(x) and F_{Y/E}(x),

for then P(E and E') = P(E)P(E'/E) = F_{X1}(x)(1-F_{Y/E}(x)+f_{Y/E}(x)).

Is this the right way of solving the problem? Is there a way of calculating f_{Y/E}(x) and F_{Y/E}(x) from f_{X1}(x), F_{X1}(x), f_{Y}(x) and F_{Y}(x)?

Any advice would be much appeciated,

Thanks

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# Pure mathematician needs help with probability!

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