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#1




Variance of the sum
Hi guys,
I want to ask the following: If X1,X2,...,Xm are iid bernoulli with q. And q is beta with a=1, b=1, and theta=1. Let S=x1+x2+...+xm what is the variance of S20? We know that Var(S20)=E(Var(s given q))+Var(E(S given q)) Now what confuses me is that should we say Var(E(S given q)) = Var(E(x1 given q)+(x2 given q)+...+(x20 given q))=Var(q+q+...+q)=Var(q)+Var(q)+..Var(q)=20*Var(q ) or Var(E(S given q)) = Var(20*q)=400 * Var(q)? When we say variance of the sum=sum of the variance and when should we square the number of terms? Any help will be appreciated, and thanks in advance!
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APC PA 
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Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com jimdaniel@actuarialseminars.com 
#3




I will attempt here to solve with more details, but I am sure I may be prone to errors, so please point them out to me.
For the case of the sum of 20 iid Bernoulli that make one binomial distribution Y with m=20: The example given of the beta distribution for q is a veiled form a continuous uniform distribution between 0 and 1. Therefore if B is subindex for Binomial and U for the Uniform As Jim Daniel pointed out the sum of the 20 Bernoulli X with q as a parameter, is a Binomial Y with m=20 and q. The expected value is The variance is Therefore
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German ______________ Prelims: VEE: Last edited by gauchodelpaso; 02012019 at 12:16 PM.. Reason: confusing  error quoting 
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It would work if each X had its own q drawn from the uniform. 
#5




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FINAL NOTE I doubt most what I had shown here, so I will delete in my prior postings the questionable or confusing parts (like Z=20X). What I found was that this has been extensively discussed related to questions in prior exams for a different beta (CFall2006, Q9 = sample SOA 253 and CSpring 2007, Q3). Basically, that we can't add variances when they are conditional in anything like q. Variance of sum is different than sum of variances. That applies only to the expected value with double expectation only (CF06/9). The sum of the variables being Bernoulli is a Binomial with m=20, as Dave pointed out. Then we apply double expectation for the variance, as I have shown (hopefully that was correct)(CS07/3).
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German ______________ Prelims: VEE: Last edited by gauchodelpaso; 02012019 at 11:38 AM.. 
#6




German,
The original post didn't say whether it was S  qthat is, the same q for all m of the Xj. [I think that this was an old exam question, and that that question said it was S  q.] With the same q for all Xj, as you said and as I posted earlier this makes S  q Binomial and the problem is straightforward from that point on. With different q's for each Xj, you would not know that the set of Xj's was mutually independent and so the problem would be impossible. Jim Quote:
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Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com jimdaniel@actuarialseminars.com 
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Suppose a variable q can be 0 or 1, 50/50. Suppose that X  q = 0 can be 1 or 2, 50/50. Suppose that X  q = 1 can be 3 or 4, 50/50. Suppose that Y  q = 0 can be 3 or 4, 50/50. Suppose that Y  q = 1 can be 1 or 2, 50/50. Suppose that, given q, X and Y are mutually independent. Compute E[XY]  E[X] E[Y] for the unconditional X and Y. It's not 0.
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Jim Daniel Jim Daniel's Actuarial Seminars www.actuarialseminars.com jimdaniel@actuarialseminars.com Last edited by Jim Daniel; 02012019 at 12:07 PM.. Reason: Clarification 
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"You are given: (i) Conditional on Q = q, the random variables X1, X2 ,…, Xm are independent and follow a Bernoulli distribution with parameter q. (ii) m 1 2 m S = X + X ++ X (iii) The distribution of Q is beta with a = 1, b = 99, and θ = 1. Determine the variance of the marginal distribution of S101." Yes, I has to be the same q for all Xi, in order to make it a binomial. From the little I read this mixture is a binomialbeta (or betabinomial) distribution in general (LM page108, example 7.12 briefly touches it). And that q has to be the same for all Bernoulli Xi as per Wikipedia: https://en.wikipedia.org/wiki/Betab...l_distribution. Quoting from Wiki: "In probability theory and statistics, the betabinomial distribution is a family of discrete probability distributions on a finite support of nonnegative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The betabinomial distribution is the binomial distribution in which the probability of success at each trial is fixed but randomly drawn from a beta distribution prior to n Bernoulli trials."" And as Academic Actuary and you pointed out, it may be confusing if it's an independent q for each Xi. Perhaps it has to do with how the model is created, like if you assign a random q for all Xi, or you draw a random q for each Xi. My knowledge is so poor that I don't know if anything like that has an intrinsic formulaic solution. Million thanks to both of you. Now, I guess we need to know if Bond (James Bond?) is satisfied with the answers...
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German ______________ Prelims: VEE: Last edited by gauchodelpaso; 02012019 at 12:46 PM.. Reason: thanking 
#10




What I am saying is assume you have an urn with different coins with the probability of heads on the coins varying according to a beta distribution. You draw 20 different coins with replacement and flip them where Xi is one if you have a head on the ith flip. The xi's are mutually independent so the variance of the sum will be of the variance of an individual flip: ( E[p(1p)] + V(p) ) x number of flips

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