<s>[INST] Rolling a die - Conditional Probability A die is thrown repeatedly.
Let $X$ ~ First 5 is thrown and $Y$ ~ First 6 is thrown
Calculate $\mathbb{E}(X|Y=3)$
You may use the identity: $\sum_{n=k}^\infty nz^{n-k} = \frac{1}{(1-z)^2}+\frac{k-1}{1-z}$
I know from the definition of expectation, we have:
$\mathbb{E}(X|Y=3) = (1*\frac{1}{5})+(2*\frac{4}{5} * \frac{1}{5}) + (3* \frac{4}{5}* \frac{4}{5} * 0) + (5* \frac{4}{5} * \frac{4}{5} * 1 * \frac{5}{6} * \frac{1}{6}) + (6* \frac{4}{5} * \frac{4}{5} * 1 * \frac{5}{6} * \frac {5}{6} * \frac{1}{6}) + ...$, where every following term, has an extra '$*\frac{5}{6}$' term and constant increases by 1.
However I am unsure of how to apply this to the identity given to find the value of the infinite sum?
 [/INST] While it is not necessary, if the question expects you to use the given identity then here is how you would go about it -
$ \small \displaystyle E(X|Y = 3) = 1 \cdot \frac{1}{5} + 2 \cdot \frac{4 \cdot 1}{5^2} + 4 \cdot \frac{4 \cdot 4}{5^2} \cdot \frac{1}{6} + 5 \cdot \frac{4 \cdot 4}{5^2} \cdot \frac{5 \cdot 1}{6^2} + ...$
$ \small \displaystyle  = \frac{13}{25} + \frac{4 \cdot 4}{5^2} \cdot \frac{1}{6} \left (4 \left(\frac{5}{6}\right)^0 + 5 \cdot \left(\frac{5}{6}\right)^1 + ...\right)$
Now note that comparing the below against the given identity,
$ \small \displaystyle 4 \left(\frac{5}{6}\right)^0 + 5 \cdot \left(\frac{5}{6}\right)^1 + ...$
$ \displaystyle \small z = \frac{5}{6}, k = 4$ and $\sum \limits_{n=k}^\infty nz^{n-k} = 54$
and we get, $ \displaystyle \small E(X|Y = 3) = \frac{157}{25}$
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