Let $$ X_1, X_2, … X_{200} \sim Ber(1/2) $$
And let $$ X = \sum_{n=1}^{200} X_n $$
I'm asked to find what is always true between these options:
A) $ P(X = 100) \le 0.6 $
B) $P(X \ge 150) < 0.5 $
C) $P(X \le 25) \le \frac{4}{7} $
D) None of the above.
The answer is C), apparently.
I don't understand how to approach this problem. How do I figure out which answer makes sense, and which doesn't? I'd appreciate any hints..
Best Answer
Even if we don't know the dependence, we still know that C) is true. Let's check the answers:
A) If all $X_1 \dots X_{100}$ are identical, and all $X_{101} \dots X_{200}$ are also identical and anticorrelated with the first subset, then $X$ is exactly $100$ so answer A is in that case not true.
B) If all $X_1 \dots X_{200}$ are identical, then $X$ is $0$ or $200$, with $P(X=0)=P(X=200)=0.5$, so $P(X>150)<0.5$ of answer B in that case is just not true.
C) Even the maximum correlation (all identical) that disproved answer B, cannot disprove C, since in that case $P(X<25)=P(X=0)=0.5$, compatible with C. Instead we use that any counter example must still have $E(X)=100$ and further that: $$\begin{align} &E(X) = E(X\ |X\leq25)\ P(X\leq 25) + E(X\ |25<X\leq 200)\ P(25<X\leq 200), \\[5pt] \text{and}\quad &E(X\ |\ X\leq 25) \leq 25\ \quad\text{(by definition, one could say) and likewise:} \\[5pt] &E(X\ |\ 25<X\leq 200) \leq 200, \\[5pt] \Rightarrow 100 = &E(X) \leq 25\ P(X\leq 25) + 200\ P(25<X\leq 200). \end{align}$$ And since $P(25<X\leq 200) = 1 - P(X\leq 25)$ it follows that $$\begin{align} 100 \leq -175\ P(X\leq 25) + 200 \\[5pt] \Rightarrow P(X\leq 25) \leq \frac{100}{175}=\frac47, \end{align}$$ so C is always true.