Exercise 3.22

Answers

(a)

Let

C_{i}

be the event that

i

-th type of coin is chosen. Let

H_{k}

be the event that

k

out of the

n

flips land heads.

P (X = k) = P (C_{1}) P (H_{k} | C_{1}) + P (C_{2}) P (H_{k} | C_{2}) = \frac{1}{2} (\binom{n}{k}) p_{1}^{k} {(1 - p_{1})}^{n - k} + \frac{1}{2} (\binom{n}{k}) p_{2}^{k} {(1 - p_{2})}^{n - k}

(b)

p_{1} = p_{2}

, then

X

is Binomial

n

k

(c)

p_{1} \neq p_{2}

, then the Bernoulli trials are not independent. If, for instance,

p_{1}

is small and

p_{2}

is large, and after the first million flips we see two heads, this increases the likelihood that we are using the coin with probability

p_{1}

of landing heads, which in turn tells us that subsequent flips are unlikely to be land heads.

fifthist

2021-12-05 00:00