Question

(a)Consider the binomial distribution with parameters. $n, \pi$ Draw 4 figures that will show qualitatively how $m_{x}, \sigma ^2_{x}$ vary with $n, \pi$

suppose we toss n = 100 unfair coins, with an unknown $\pi$

(b) what is the expected number of heads out of n? (the answer depends on $\pi$ )

(c) what is the typical deviation of the number of heads out of n? (the answer depends on $\pi$ )

(d) what is the largest typical deviation of the number of heads out of n? (the answer is a number) Hint: consult your graph of $\sigma^2$ vs $\pi$ in part a)

Answer 1

a)

$\mu_x$ = n $\pi$

first and second graph will increase linearly keeping the other parameter constant

$\sigma ^2_x = n \pi(1-\pi )$

third graph also linear with respect to n

fourth graph will be inverted parabola , maximum at pi = 1/2 and 0 when pi = 0 and 1

b)

$\mu_x$ = n $\pi$

c)

$\sigma ^2_x = n \pi(1-\pi )$

$\sigma_x$ =sqrt(n $\pi$ (1-$\pi$))

d)

$\sigma ^2_x = n \pi(1-\pi )$

largest value when $\pi$ = 1/2

$\sigma_x$ =sqrt(n/4)

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