Question

A set of 7 fair coins was tossed 259 times and the frequency of throws

observed were as follows:

Number of Heads : 0 1 2 3 4 5 6 7

Frequency of throws : 3 7 23 62 63 53 37 11

Fit a binomial distribution and find mean and standard deviation of fitted

distribution.

Answer 1

mean:

$\mu=\sum x_if_i/N=4.07$

$p=\mu/n=4.07/7=0.58$

$q=1-p=0.42$

$N=259$

$n=7$

$f(x=k)=NC^k_np^kq^{n-k}$

$f(x=0)=259q^{7}=0.6\approx1$

$f(x=1)=259\cdot7pq^{6}=5.77\approx6$

$f(x=2)=259\cdot C^2_7p^2q^{5}=23.91\approx24$

$f(x=3)=259\cdot C^3_7p^3q^{4}=55.03\approx55$

$f(x=4)=259\cdot C^4_7p^4q^{3}=76$

$f(x=5)=259\cdot C^5_7p^5q^{2}=62.97\approx63$

$f(x=6)=259\cdot7p^6q=28.99\approx29$

$f(x=7)=259p^7=5.71\approx6$

for fitted data:

mean:

$\mu=\sum x_if_i/N=4.05$

standard deviation:

$\sigma=\sqrt{\frac{\sum f_ix_i-(\sum f_i x_i)^2/N}{N}}=1.32$