Question

sing R please, 1. Generate 5,000 iid samples from the standard normal distribution; compute the mean of these random samples. Repeat this process 100 times so you would have stored 100 sample averages. Plot the sample averages using hist() and lines(density() in R. What do you observe? 2. Compute the mean and variance of Binomial distribution with parameters m and p directly from definition 3. Given a random sample of size n from a Poisson population, find the maximum likelihood estimator for Assume now that the sample size is 1,000, generate such a sample in R and plot the likelihood function. Explain what you see.

please i need to solve 2 and 3 and please explain everything and write I commend

Answer 1

Q2)

from definition 922 LXNBCmie) 2 a 12 an pix=n) = m pa di-pm-x from m defenition mean = E(0) - E x p(x)= E a.m x=0 Ela) = px (1-pom- 250 mn mb 3 X2 Now, 24 اره و mp. ( a + pn-1 = mb & where q = itp Now Now, ELX2) = n2/my pro m 12 220 1-2 E E {acu-1 tay mcm-1). mm-1 1 2 - 3 WzD X - 2 my Le no mom-12p² e 1-2 - + mp Lx= 22-2 a mim. 1) p² (q + p/m- 2 tmp = mcm-1) p² + mp so variance vex) = ECx²) ECA - м с м 21 p +mp - м-р- 2 m² m2 + mp-m2 mp-mp²= mpcl-p) mba

Q3) Please ignore Q2) written in the image this is your answer for question 3.

292) P(X=2) = e-^ a) likelihood an L = echt aber I di t=1 log - likelihood log i = -ux & & ni log A-log Ti si b) de landele cu tome - का c) put allogl=0 - u teniso

R code for part 3

llh=rep(0,40)

#different paramenters to see which one we maximise
lambdas=seq(1,20.5,by=0.5)
for(i in 1:40)
{ #here we are generating random sample of 1000 from poisson with parameter 6
y=rpois(1000,6)
#llh is log likelihood
llh[i]<- sum(dpois(y,lambdas[i], log=TRUE))
}
final=data.frame(lambdas,llh)
plot(final)

Run 2 Source = H E Import Dataset, = List Source on Save 1 lambda=2 2 llh=rep(0,40) (Global Environment - values 40L lambda lambdas llh 4 #different paramenters to see which one we maximise 5 lambdas=seq(1,20.5, by=0.5), 6 for(i in 1:40) 7. { #here we are generating random sample of 1000 from poisson with parameter y=rpois(1000,6) 9 #Llh is Log likelihood 10 11h[i]<- sum(dpois(y, lambdas[i], log=TRUE)) 11 } 12 final=data.frame (lambdas, 11h), 13 plot(final) num [1:40] 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 ... num [1:40] -7965 -6175 -4879 -4085 -3426 ... int (1:1000] 10 5 3 6 5 76 774 ... Help Viewer Export- Publishe Files Plots Packages Zoom OOOOOO 14 15 -3000 -5000 411 -7000 15:1 (Top Level) = R Script = Console Terminal IV.CC 0006- 38 19.5 -8632.765 39 20.0 -9058.220 40 20.5 -9489.438 > plot(final) un - lambdas

So we can in the plot we get our maximum at lambda=6 from which we generated the poisson sample hence validating our experiment.

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