Help with exponential distribution

[sneaky666]

X1,X2,...,XN are independently identically exponentially distributed with expected value of 5. How can I compute X[bar]n when n=20 and N=1000? Then compute the proportion of values of X[bar]n that lie between 6.99 and 7.01.

repeat the above question with n=100


My thoughts

so basically i am using code in R software to do this
and basically this question means that suppose there are 1000 iid exp. dist. with Expected value of each X is 5. Then get the mean of each X where the number of observations is 20 then 100.
i used
a=(1:1000) <--makes a vector [1,2,3,...,1000]
for(i in a){a=mean(rexp(20,5))} <---so 20 observations and expected value of 5, each slot in the vector gets replaced by a mean
plot(a)
and
a=(1:1000)
for(i in a){a=mean(rexp(100,5))}
plot(a)
i get the y-axis to be 0.1 to 0.35 for n=20 and 0.14 to 0.26 for n=100, what am I doing wrong? I don't get how to get a proportion of values of X[bar]n that lie between 6.99 and 7.01.

 

[AlephZero]

and basically this question means that suppose there are 1000 iid exp. dist. with Expected value of each X is 5.

Correct, and before you start writing any code, think about what that statement implies.

For an exponential distribution, if you are told the mean you also know the variance.

What general results do you know about the distribution of the sum of independent identically distributed variables?