A biased coin with probability q of landing heads is repeatedly tossed until the first head isseen. The number of tails X before the first head is modelled as a geometric distribution withprobability mass function P(X = x) = g(1-q). The experiment was repeated n times andx1,x2,..., Xn tails were observed.(a) Write down the likelihood for q. Show that the maximum likelihood estimate for q isâ =nn+Snwhere S = x₁.(b) Find the Fisher information and hence the asymptotic variance for â.(c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that theposterior distribution is Beta(a₁,₁), where you should determine a₁ and ₁.(d) We have n = 5 and observed data x₁,...,x₂ = 4,2,5,6,3.(i) What is the maximum likelihood estimate â?(ii) Find an approximate 95% confidence interval for q.(iii) Before seeing the data, our probability distribution for q has mean 0.4 andstandard deviation 0.2. Find values of ao and Bo corresponding to this belief. Whatis then the posterior distribution for q? What is the posterior mean?(iv) Comment on the posterior mean compared to the maximum likelihood estimateand the prior mean for this example. No further calculations or formulae areneeded here.

As per the Bartleby guildlines we have to solve first three subparts and rest can be reposted....…

A biased coin with probability q of landing heads is repeatedly tossed until the first head is seen. The number of tails X before the first head is modelled as a geometric distribution with probability mass function P(X = x) = q(1-q)*. The experiment was repeated n times and x1,x2,...,xn tails were observed. (a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is â= n n+S where S = xj. (b) Find the Fisher information and hence the asymptotic variance for â. (c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the posterior distribution is Beta(a₁,31), where you should determine a₁ and ₁.

A biased coin with probability q of landing heads is repeatedly tossed until the first head is seen. The number of tails X before the first head is modelled as a geometric distribution with probability mass function P(X = x) = q(1-q)*. The experiment was repeated n times and x1,x2,...,xn tails were observed. (a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is â= n n+S where S = xj. (b) Find the Fisher information and hence the asymptotic variance for â. (c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the posterior distribution is Beta(a₁,31), where you should determine a₁ and ₁.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

A biased coin with probability q of landing heads is repeatedly tossed until the first head is
seen. The number of tails X before the first head is modelled as a geometric distribution with
probability mass function P(X = x) = g(1-q). The experiment was repeated n times and
x1,x2,..., Xn tails were observed.
(a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is
â =
n
n+S
n
where S = x₁.
(b) Find the Fisher information and hence the asymptotic variance for â.
(c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the
posterior distribution is Beta(a₁,₁), where you should determine a₁ and ₁.
(d) We have n = 5 and observed data x₁,...,x₂ = 4,2,5,6,3.
(i) What is the maximum likelihood estimate â?
(ii) Find an approximate 95% confidence interval for q.
(iii) Before seeing the data, our probability distribution for q has mean 0.4 and
standard deviation 0.2. Find values of ao and Bo corresponding to this belief. What
is then the posterior distribution for q? What is the posterior mean?
(iv) Comment on the posterior mean compared to the maximum likelihood estimate
and the prior mean for this example. No further calculations or formulae are
needed here.

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