1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability, distributions

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data

    We have an interval [0,1], which we divide into k equally sized subintervals and generate n observations. Lets call the number of observations which falls into interval k_i, X_i. What distribution does X_1 have?

    Now we define Y_i=X_i/n. Derive the Expected value, variance and standard deviation for Y_i?

    This is a homework assignment, so plz just guide me... dont give me the answers :)

    2. Relevant equations


    3. The attempt at a solution

    The distribution for X_1: The amount of observations in each interval should follow a normal distribution, no? But the number of observations in each interval will be discrete? If I could understand what distr. this is, I could solve for E(X_i^2) in the last expression?


    X_i=# of n that is in k_i. So, E(X_i)=n/k.
    E(Y_i)=E(X_i/n)=(1/n)(E(X_i))=1/k
    V(Y_i)=E((Y_i)^2)-(E(Y_i))^2=E((X_i/n)^2)-(1/k)^2=(1/n)^2*E(X_i^2)-1/k^2 ?????
     
  2. jcsd
  3. Feb 7, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    How do you pick a random point in [0,1]? Do you use a uniform distribution? If so, then NO, the distribution of X_1 is not normal. In fact, for ANY distribution on [0,1], the distribution of X_1 is not normal: the normal distribution goes from -∞ to +∞, but the distribution of X_1 only goes from 0 to n. Of course the number of points in each interval will be discrete; after all, you just pick an integer number n of points altogether, and the number falling into an interval will be some integer from 0 to n.

    To understand what is the distribution of X_1, you first need to say what is the distribution of the random points on [0,1]. If it IS the uniform distribution, draw a diagram of its density function f(x), and remember what "density" means (or look it up in a book or a web page).
     
    Last edited: Feb 7, 2013
  4. Feb 7, 2013 #3
    Hi Ray, thx for your answer.

    You are correct, I forgot the "small" little detail that we generated it from a uniform distribution. I think/thought I knew what a density function is, and that the probability function for a uniform distribution is 1/(b-a+1). I cant udnerstand what the distribution for X_1 will be tho. Let's say we generate 100 numbers in [0,1] and create 10 subintervals with a uniform distr. Our E(X_i)=10, so for i=1,...,10 we will have a rectangular shaped diagram, the usual uniform one. If I picture a diagram, with X_1 on the Y-axis and the x-axis goes from [0,1/10], so it will only vary discretly in the y-axis around 10. This will make it a discretly uniform distr.?

    excuse my english, I dont know some of the terms in english.
     
  5. Feb 7, 2013 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    What is the probability that the first generated point lies in the interval [0,1/10]? What is the probability that the second generated point lies in the interval [0,1/10]? Just continue like that.
     
  6. Feb 7, 2013 #5
    (1/10)^n?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Probability, distributions
  1. Probability Distribution (Replies: 24)

  2. Probability Distribution (Replies: 13)

Loading...