i am stuck on this question and don't know how to solve it. would really appreciate your help with it:
we toss a regular coin 50 times:
$X$ - the number of heads from 50 tosses.
$Y$ - the number of heads from the first 20 tosses.
1) find the conditional probability function of Y if given that X=i for {i=0,...,50}, and the conditional probability function of X if given Y=j so that {j=0,...,20}
2) calculate $E[X\mid Y =j]$
what i tried to do:
1)in order to find the conditional probability function, we have to use $P_{X|Y}(x \mid y) =\frac{P_{XY}{(x,y)}}{p_Y(y)}$, however, since it's cumulative it becomes:$ P_{X\mid Y}(x \mid y) = P(X ≤ x \mid Y = y)$, which means that in order to find the conditional probability function of Y if given that X=i, we have to calculate $P_{X\mid Y}(x\mid y)=P(X≤50\mid Y=y)$, and to calculate the other (the conditional probability function of X if given Y=j so that {j=0,...,20}) we have to calculate $P_{X\mid Y}(x\mid y)=P(Y≤20\mid X=X)$. however i'm not sure how to continue the calculation to find the needed conditional probability functions
2)it seems easier: in order to find $E[X\mid Y =j]$, i have to calculate $\sum_x x p_{(X\mid Y)}(x\mid y)$ which is actually equivalent to using the integral $\int x f_{x\mid y} (x\mid y)$. however, i don't know how to calculate $f_{x\mid y}(x\mid y)$ to be able to calculate the conditional mean.
could you please help me with that or correct me if what i did is wrong?
thank you very much, tried to elaborate as much as i can. would appreciate your help learning how to calculate it correctly
Tip: The count of heads among the first 20 tosses and the count of heads among the last 30 are independent random variables. Also both are binomially distributed, although not identically.
So, let $Z=X-Y$, then $P_{X,Y}(x,y)=P_{Z}(x-y)\cdot P_Y(y)$
Therefore ...
For (b) $\mathsf E(X\mid Y=j)=\mathsf E(Z+Y\mid Y=j)$, so...