could you please do part a C2. (a) (i) Suppose that events B₁, B2, ..., Bk partition the sample space , and that eventA is some other event.Bayes' Theorem states thatPr(B; A)Derive Bayes' Theorem, clearly stating any necessary probability definitions andclearly stating the conditions which make the events B₁, B2, ..., Bk partition thesample space.Pr(A|B₂)Pr(B₂)Pr(A)2i = 1, 2..., k.(ii) Chronic heart disease (CHD) effects about 5% of the adult population. Anelectrocardiogram (ECG) stress test is 60% effective at detecting CHD when itis present but also suggests CHD in 10% of cases where it is not present.Find the probability that a randomly selected adult from the population has apositive test result?Suppose now that a particular patient does have a positive test, then what is theprobability that the individual actually has chronic heart disease?(b) A particular manufacturer of mobile phone battery claims a life of 20 hours beforeneeding recharging. To investigate this claim, randomly selected fully-charged bat-teries are used until they go flat. A random variable X is used to indicate successor failure. If the recorded time is at least 20 hours then X = 1, otherwise X = 0,and let Pr(X = 1) = p. Suppose that five batteries are tested giving the observedsequence x₁=1, x2 = 0, x3 = 1, x4 = 1, x5 = 1.(i) Explain why a Bernoulli distribution would be a suitable model for this situation.(ii) As part of a Bayesian model, explain why a beta distribution would be a suitableprior model for p, that is p Beta (a, b).(iii) Write down the appropriate form of Bayes' Theorem which defines the posteriordistribution, f(px), in terms of the data likelihood, 1(xp), the prior distribution,f(p), and the data evidence f(x).Using the result that after the first battery is tested, the posterior distribution ofpx is Beta(a + x₁, ß + 1 - x₁) and that expert opinion suggests using a = 10and 3 = 1, then what is the posterior mean?(iv) Explain how the posterior distribution and posterior mean in (b)(iii) would changeas the test results from the other batteries are included sequentially.Hint: You may use, without proof, that for a beta distribution, Z Beta (a, b) themean is E[Z] = a/(a + b) and the variance is Var[Z] = ab/{(a + b)²(a+b+1)}.

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Note: As the student has asked to answer part A, answer has been provided for Part A only.