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45.18 Suppose that X[- Xz- arciid_ Bemoulli() whcre unknown_ Delerminc lower bound on n s0 that the probability that the estimate M; will be within of thc unknown v...

Question

45.18 Suppose that X[- Xz- arciid_ Bemoulli() whcre unknown_ Delerminc lower bound on n s0 that the probability that the estimate M; will be within of thc unknown valuc of is about 0.9974. This allows us t0 rn simulations with high confidence that the crror in the approximation quoted_ 1s less than somc prescribed valuc (Hint: Usc thc fact that x (4 Tr) < 4/4 for all x [0. 1].)

45.18 Suppose that X[- Xz- arciid_ Bemoulli() whcre unknown_ Delerminc lower bound on n s0 that the probability that the estimate M; will be within of thc unknown valuc of is about 0.9974. This allows us t0 rn simulations with high confidence that the crror in the approximation quoted_ 1s less than somc prescribed valuc (Hint: Usc thc fact that x (4 Tr) < 4/4 for all x [0. 1].)



Answers

Refer to Exercise $3.86 .$ The maximum likelihood estimator for $p$ is $1 / Y$ (note that $Y$ is the geometric random variable, not a particular value of it). Derive $E(1 / Y) .[\text { Hint: If }|r|<1$, $\left.\sum_{i=1}^{\infty} r^{i} / i=-\ln (1-r) .\right]$.

Hi here our problem is to find that we have been given X one extra so on eggs in these are the random samples which follows personally distribution with probability of success be okay. And it is also given that piece restricted says that how less than equals to be less than it costs to what? No we have to find the Emily of B. Okay now first we first when we write the poverty mass function then we we will take the usual parametric space of B which is zero less than be less than one. Okay now the property mass function of this random variable is P. X equals two probability capital X equals two small X. Because to B to the power eggs 1 -2 to the power one minus X says that X equals to zero comma one. It takes only two values and zero less than be less than one. Okay now to find family of me we first construct the likelihood function which is united by L. B. Ex school equals two productive irons from one to end B. Exactly this is we do the parcel mentioning side 1 -4. to the power In- Submission Exile Islands. From one point in all cases. Okay, now maximizing this like look function is equivalent to maximizing the log likelihood function. Now we write log leg blue function as scribble the Kahlan excursion which is equals two submission exile lot B plus in minus the mission next day. No one minus and the family of the way first. And the first started derivative of this script log likelihood function And equated to zero. Okay. No first order derivative with respect to be script. L. P. X. Girl Equals to zero. That implies submission excited liberated by P. Plus in my estimation next day Only baited by one -P. And again -1 coach to syria so we can ride some issue. Makes minus someone should excite Into B equals two N. P minus pe submission. Except this is really empty. Okay now clearly Submission X. i. equals two MP. So we had Emily calls to submission excited by and the many of me Culture to submission. Excite islands from one doing for liberated by. And this is the value of P. When he is not restricted. See the maximum likelihood estimated of pes expert. Okay now Excite next of value zero and 1. Okay so expert definitely will lie between zero less than equals to expire less than equals two facebook users. Now we do you hide these interval in two intervals. Okay zero lesbian Eggs. two eggs were less than half and also half less than equals two. Expert less them. Just one. Now when if he is restricted half less than nickels to be less than equals to one. Then definitely we had Emily Takes two values. What are those two values when zero is done equals two X. Bar less than equals to have and another is sorry it will be only less than And another is half less than equals two. Expert. Less than industry. One when half less than the constraints for less than one. So definitely be Emily of people be expert. Mm. And when the expo lies between zero to have the definitely takes the value only. Have. Yeah. So what we get half when X body is less than half and expire when eggs, berries catered, then calls to This is in college too. Maximum of how comma. Thanks. So when we is restricted, Emily of, we had Emily of me wants to be had maxie mama. How coma explosion. It's one of the brands. Thank you.


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