- #1
Eclair_de_XII
- 1,083
- 91
- Homework Statement
- "Let ##Y_i## be independent Bernoulli r.v.'s. Let ##T=1## if ##Y_1=1## and ##Y_2=0## and ##T=0## otherwise. Let ##W=\sum Y_i##. Prove that ##P(T=1|W=w)=\frac{w(n-w)}{n(n-1)}##.
- Relevant Equations
- ##P(A|B)=\frac{P(A\cap B)}{P(B)}##
An answer from another part in the problem: ##E(T)=p(1-p)##
##P(T=1|W=w)=\frac{P(\{T=1\}\cap\{W=w\})}{P(W=w)}=\frac{\binom {n-2} {w-1} p^{w-1}(1-p)^{(n-2)-(w-1)}}{\binom n w p^w (1-p)^{n-w}}=\frac{(n-2)!}{(w-1)!(n-w-1)!}\frac{w!(n-w)!}{n!}\frac{1}{p(1-p)}=\frac{w(n-w)}{n(n-1)}(p(1-p))^{-1}##.
I cannot seem to get the terms with ##p## out of my expression.
I cannot seem to get the terms with ##p## out of my expression.