+ Reply to Thread
Results 1 to 8 of 8

Thread: help with exponential distribution

  1. #1
    Points: 1,899, Level: 25
    Level completed: 99%, Points required for next Level: 1

    Posts
    49
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Exclamation help with exponential distribution




    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[i]=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[i]=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 dont get how to get a proportion of values of X[bar]n that lie between 6.99 and 7.01.

  2. #2
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: help with exponential distribution

    Quote Originally Posted by omega View Post
    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.
    ..... I dont get how to get a proportion of values of X[bar]n that lie between 6.99 and 7.01.
    Obviously, one way of doing this (obtain the proportion) would be to use a normal approximation...Is this the intent of the assignment you're working on?...Or is it something else?

  3. #3
    Points: 1,899, Level: 25
    Level completed: 99%, Points required for next Level: 1

    Posts
    49
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: help with exponential distribution

    It didn't say anything about using normal approximation but how would you do it if I had to?

  4. #4
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: help with exponential distribution

    Quote Originally Posted by omega View Post
    It didn't say anything about using normal approximation but how would you do it if I had to?
    It's easy because we know that the mean is 5 and therefore the standard deviation will also be 5.

    Thus, we have

    6.99 = z*(5/Sqrt[n]) + 5

    7.01 = z*(5/Sqrt[n]) + 5.

    Just solve for z in both cases and use the normal curve to determine the proportions between the two points.

  5. #5
    Points: 1,899, Level: 25
    Level completed: 99%, Points required for next Level: 1

    Posts
    49
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: help with exponential distribution

    so the formula is
    x = (mu*z) / sqrt(n) + mu

    so then i get

    (x-mu)sqrt(n) / mu = z

    so

    (6.99-5)sqrt(20) / 5 = z
    z =~1.77991011
    (7.01-5)sqrt(20) / 5 = z
    z =~1.797795654

    (6.99-5)sqrt(100) / 5 = z
    z =3.98
    (7.01-5)sqrt(100) / 5 = z
    z =4.02

    So here i just get the area under the curve between the two points for each one, from the *standard normal distribution* ?

  6. #6
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: help with exponential distribution

    Quote Originally Posted by omega View Post

    So here i just get the area under the curve between the two points for each one, from the *standard normal distribution* ?
    Yes, the standard normal distribution.

  7. #7
    Points: 1,899, Level: 25
    Level completed: 99%, Points required for next Level: 1

    Posts
    49
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: help with exponential distribution

    so for n=20 i get 0.0014406382
    and for n=100 i get 0.0000053585


    Does this seem right?

  8. #8
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: help with exponential distribution


    Quote Originally Posted by omega View Post
    so for n=20 i get 0.0014406382
    and for n=100 i get 0.0000053585


    Does this seem right?
    Yes, I checked your calculation for n=100, and it is correct.

    However, note that this is a normal approximation to the distribution of the sample means.

    I also checked empirically (based on n=100 columns with 1000000 rows) for n=100 and I obtained a proportion of 0.00065. So the normal approximation is going to be a bit off.

    The reason is because the sampling distribution of the mean will still be somewhat positively skewed and have some kurtosis. More precisely, the skew of the sampling distribution will be 2/Sqrt[n] and the kurtosis will be 6/n.

+ Reply to Thread

           




Similar Threads

  1. Exponential Distribution
    By jwill212 in forum Statistics
    Replies: 1
    Last Post: 03-02-2011, 07:57 PM
  2. exponential distribution
    By trbbls in forum Statistics
    Replies: 1
    Last Post: 03-17-2008, 09:16 PM
  3. Replies: 0
    Last Post: 01-23-2008, 11:32 PM
  4. Exponential Distribution
    By jaknowlden in forum Probability
    Replies: 7
    Last Post: 10-26-2006, 07:53 PM
  5. exponential distribution
    By skip_it2 in forum General Discussion
    Replies: 0
    Last Post: 03-08-2006, 08:50 PM

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats