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gccdman
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Hi,
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 X1 and X2 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:
X1 <= a <= X1+X2
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 X1 and Y= X1 + X2
X1: S -> R is defined by th identity function X1(s) = s.
Hence, f1(x) = P(X1 = x) = P({s € S: X1(s) = x}) = P({s€S: s = x) = 0 if x does not belong to {0,2,...,2n}; 1/(n+1) if x € S
Since X2: S -> R is defined by X2(s) = s we have f1(x) = f2(x).
Let Y = X1 + X2. Y: SxS -> R is defined by Y(s,s') = X1(s) + X2(s') = s + s'.
Hence fY(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 X1 and Y = X1 + X2 which I shall denote FX1(x) and FY(x) respectively.
Let E be the event X1 <= x, E' the event x <= Y.
Now if E and E' were independant by definiton P(E and E') = P(E)P(E') = FX1(x)(1-FY(x)+fY(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 fY/E(x) and FY/E(x),
for then P(E and E') = P(E)P(E'/E) = FX1(x)(1-FY/E(x)+fY/E(x)).
Is this the right way of solving the problem? Is there a way of calculating fY/E(x) and FY/E(x) from fX1(x), FX1(x), fY(x) and FY(x)?
Any advice would be much appeciated,
Thanks
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 X1 and X2 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:
X1 <= a <= X1+X2
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 X1 and Y= X1 + X2
X1: S -> R is defined by th identity function X1(s) = s.
Hence, f1(x) = P(X1 = x) = P({s € S: X1(s) = x}) = P({s€S: s = x) = 0 if x does not belong to {0,2,...,2n}; 1/(n+1) if x € S
Since X2: S -> R is defined by X2(s) = s we have f1(x) = f2(x).
Let Y = X1 + X2. Y: SxS -> R is defined by Y(s,s') = X1(s) + X2(s') = s + s'.
Hence fY(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 X1 and Y = X1 + X2 which I shall denote FX1(x) and FY(x) respectively.
Let E be the event X1 <= x, E' the event x <= Y.
Now if E and E' were independant by definiton P(E and E') = P(E)P(E') = FX1(x)(1-FY(x)+fY(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 fY/E(x) and FY/E(x),
for then P(E and E') = P(E)P(E'/E) = FX1(x)(1-FY/E(x)+fY/E(x)).
Is this the right way of solving the problem? Is there a way of calculating fY/E(x) and FY/E(x) from fX1(x), FX1(x), fY(x) and FY(x)?
Any advice would be much appeciated,
Thanks
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