Exercise 3.44

Answers

(a)

P (X \oplus Y) \sim

Bern

(\frac{p}{2})

(b)

p \neq \frac{1}{2}

X \oplus Y

and

Y

are not independent. Imagine that

X = 0

is extremely unlikely. Then, knowing that

Y = 0

makes it very likely that

X \oplus Y = 1

. If

p = \frac{1}{2}

, then

X \oplus Y

and

Y

are independent.

$X \oplus Y$ and $X$ are independent, since knowledge of $X$ still keeps the probability of $Y = 1$ at $\frac{1}{2}$

(c)

Let the largest element in $J$ be $m$ .

\begin{array}{l} P (Y_{J} = 1) & = P (X_{m} = 1) P (Y_{J ∖ {m}} = 0) + P (X_{m} = 0) P (Y_{J ∖ {m}} = 1) \\ = \frac{1}{2} (P (Y_{J ∖ {m}} = 0) + P (Y_{J ∖ {m}} = 1)) \\ = \frac{1}{2} \end{array}

Thus, $Y_{J} \sim Bern (\frac{1}{2})$

fifthist

2021-12-05 00:00