Question

Preview of Homework 1 R.1 The probability distribution for diff R.2 The probability distribution for max R.3 Expected value of diff R.4 Expected value of max R.6 Expected diff squared R.7 Expected max squared R.9 Variance of diff R.10 Variance of max Histogram for max Histogram for diff

Need help with the codings in R Markdown (in R Studio)

Answer 1

Sol: Since sample mean and sample variance is unbiased estimator of population mean and population variance respectively.

$E(\bar{x})=\mu$    and

$E(s^2)=\sigma ^2$

Where,

$\bar{x}$ is a sample mean and $\mu$ is a population mean.

$s^2=\frac{1}{n-1}\sum (x_{i}-\bar{x})^2$ is sample variance.

$\sigma ^2=\frac{1}{n}\sum (x_{i}-\bar{x})^2$ is population variance.

Therefore, R-code result gives approximately same value of sample variance (dividing by n-1) and population variance (dividing by n)

> red=sample(c(1:6),repl=T,size=10000)
> green=sample(c(1:6),repl=T,size=10000)
> sum=red+green
> max=ifelse(red>green, red, green)
> diff=abs(red-green)
> table(sum)
sum
   2    3    4    5    6    7    8    9   10   11   12
270 555 850 1132 1380 1649 1372 1084 848 591 269
> mean(sum)
[1] 7.0029
> mean(sum^2)
[1] 54.9115
> var(sum)
[1] 5.871479
> sd(sum)
[1] 2.423113
> sum((sum-mean(sum))^2)/9999 # This case provides sample variance.
[1] 5.871479
> mean(sum^2)-mean(sum)^2    # This case provides population variance.
[1] 5.870892
> hist(sum)