The following is the R code (all statements starting with # are comments)
get this plot
We can see that as the sample size increases, the dotted vertical line moves closer to the solid line corresponding to , showing the asymptotic unbiasedness
We can also observe that the variance of the sampling distribution reduces with the increase in sample size.
Finally we can also see that the density takes on a bell shape as the sample size increases, indicating the asymptotic normality.
The code in text format is below
------------------------------------
#set the mean
mu<-10
# set the standard deviation
sigma<-8
#set the random seed
set.seed(123)
#make an empty plot to add the curves later
plot(1, type="n", xlab=bquote(bar(X)^2), ylab="Density", xlim=c(0,
800),ylim=c(0,0.02),
main=bquote("The distribution of "*bar(X)^2))
#set the colors for the plot
cols<-c("red","blue","green","magenta")
#set the sample sizes
n<-c(3,6,10,50)
#do this for each sample size n
for (i in 1:length(n)){
#set the number of samples to be taken
r<-50000
#draw a sample of size n, r times, that is n*r draws
from normal(10,8^2)
x<-rnorm(n[i]*r,mu,sigma)
#make this into a matrix of n rows and r columns
x<-matrix(x,nrow=n[i],ncol=r)
#get r sample means
xbar<-apply(x,2,mean)
#get the square of means
xbar2<-xbar^2
#draw the density of xbar2
lines(density(xbar2),typ="l",col=cols[i])
#add the vertical line
abline(v=mean(xbar2),col=cols[i],lty=2)
}
#add a solid line for mu^2
abline(v=mu^2,lwd=2)
#add the legend
legend("topright",paste("n=",n,sep=""),col=cols,lty=1)
R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which...
R2 Let X ~ N(μ = 10.82). Following up on RI, we will be approximating μ2, which we can see should be 100. For now, let the sample size be n 3, Pick 3 random numbers from X, compute 72 X, and repeat the process a total of 50000 times. Plot a smooth version of the histogram of these 50000 values for X2: the plot(density(...). command in R will be useful. Now find the average of your 50000 values and...
R2 IS THE QUESTION,THANKS! R2. Let X ~ N(μ = 10.82). Following upon we be approximating μwe can see should be 100. For now, let the sample size be n = 3, Pick 3 random numbers from X, compute X', and repeat the process a total of 50000 times. Plot a smooth version of the histogram of these 50000 values for X: the plot(density(...)) command in R will be useful. Now find the average of your 50000 values and make...
R1. Suppose X is a continuous RV with E(X-μ and Var(X-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ2. (This problem does not require the computer.) R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which we can see should be 100, For now, let the sample size be n 3. Pick 3 random numbers from...
RI. Suppose X is a continuous RV with E(X)-μ and Var(X)-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ. (This problem does not require the computer.) R2. Let X ~ ŅĢi-10.82). Following up on RI, we will be approximating μ2, which we can see should be 100. For now, let the sample size be n = 3, Pick 3 random numbers from...
We start out with a couple of defintions and examples. Definition: Let X and Y have joint pdf f(x,y). The conditional pdf of Y given X = x (resp. of X given Y = y) is defined by h(y|x) = f (x, y) resp. g(x|y) = f (x, y) f1(x) f2(y) If A is a subset of the real line, then P(Y ∈A|X =x)= h(y|x)dy resp. P(X ∈A|Y =y)= g(x|y)dx . AA Example 1 (seen in class) Consider the joint...
A polynomial p(x) is an expression in variable x which is in the form axn + bxn-1 + …. + jx + k, where a, b, …, j, k are real numbers, and n is a non-negative integer. n is called the degree of polynomial. Every term in a polynomial consists of a coefficient and an exponent. For example, for the first term axn, a is the coefficient and n is the exponent. This assignment is about representing and computing...
Experimental technique 5. [1pt] Which of the following statements are true for this experiment? E.g., enter AB. If none are true, enter N. You have 4 tries. Everyone must wear safety glasses. For each launch angle, the projectile range is measured for one speed setting. Never look into the barrel of the launcher when it is loaded. The launch angle θ measured using the plumb line is the angle of the launcher with respect to the vertical. The projectile is...
Need help with stats true or false questions Decide (with short explanations) whether the following statements are true or false a) We consider the model y-Ao +A(z) +E. Let (-0.01, 1.5) be a 95% confidence interval for A In this case, a t-test with significance level 1% rejects the null hypothesis Ho : A-0 against a two sided alternative. b) Complicated models with a lot of parameters are better for prediction then simple models with just a few parameters c)...
1. For each of the following regression models, write down the X matrix and 3 vector. Assume in both cases that there are four observations (a) Y BoB1X1 + B2X1X2 (b) log Y Bo B1XiB2X2+ 2. For each of the following regression models, write down the X matrix and vector. Assume in both cases that there are five observations. (a) YB1XB2X2+BXE (b) VYBoB, X,a +2 log10 X2+E regression model never reduces R2, why 3. If adding predictor variables to a...
Partner: Date Name 11 Snell's Law Introduction When light passes from one material to another it is always bent away from its original path. This process is known as refraction and the change in direction depends on the change in optical density (or refractive index) of the two materials. A larger change in refractive index results in a larger change in angle between incoming and outgoing light beams. A light beam bends closer to the normal in the material with...