2022-03-22
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asked 2022-04-22

Given you have an independent random sample X1,X2,,Xn of a Bernoulli random variable with parameter p, estimate the variance of the maximum likelihood estimator of p using the Cramer-Rao lower bound for the variance
So, with large enough sample size, I know the population mean of the estimator P^ will be p, and the variance will be:
Var[P^]=1nE[((/p) lnfx(X))2]
Now I'm having some trouble calculating the variance of P^, this is what I have so far:
since the probability function of X is binomial, we have:
fx(X¯)=ni=1nXi*pi=1nXi*(1-p)n-i=1nXi
so: lnfX(X)=ln(ni=1nXi)+i=1nXiln(p)+(n-i=1nXi)ln(1-p)
and: lnfX(X)p=i=1nXip-(n-i=1nXi)(1-p)=nX¯p-(n-nX¯)(1-p)
and: (lnfX(X)p)2=(nX¯p-(n-nX¯)(1-p))2=n2p2-2n2pX¯+n2X¯2p2(1-p)2
since E[X2]=μ2+σ2n, and for a Bernoulli random variable E[X]=μ=p=E[X] and Var[X]=σ2=p(1p):
E[(ln:{fX(X)}p)2]=n2p22n2pE[X]+n2E[X2]p2(1p)2=n2p22n2p2+n2(p2+p(1p)n)p2(1p)2=np(1p)p2(1p)2=np(1p)
Therefore, Var[P^]=1nE[((/p) lnfx(X))2]=1nnp(1-p)=p(1-p)n2
However, I believe the true value I should have come up with is p(1p)n.

asked 2022-04-08

find the 55th term in the following arithmetic sequence: -102, -98, -94, -90

 

asked 2022-03-04

 Brand of Smartphone  Male A43B34C48
A. What is the value for the degrees of freedom?
B. If the expected number of males with smartphone brand B is 32.5, what is the chi-square component?
C. If the test statistic (chi-square) = 6.40, then the p-value =
D. If the significance level was 0.05, should the researcher reject or fail to reject the null?

asked 2022-04-20

find all values of k sor which they given matrix augmented matrix to a consistent linear system

asked 2022-03-29

Find a vector function that represents the curve of intersection of the two surfaces of the cylinder x^2+y^2=4 and the surface z=xy.

asked 2022-03-27

 

I need solution of Q2 of given assignment

asked 2022-04-19

J. P. Morgan Asset Management publishes information about financial investments. Between 2002 and 2011 the expected return for the S&P was with a standard deviation of and the expected return over that same period for a Core Bonds fund was with a standard deviation of (J. P. Morgan Asset Management, Guide to the Markets). The publication also reported that the correlation between the S&P and Core Bonds is . You are considering portfolio investments that are composed of an S&P index fund and a Core Bonds fund. a.  Using the information provided, determine the covariance between the S&P and Core Bonds. Round your answer to two decimal places. If required enter negative values as negative numbers.


 


 

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