What Distribution Does X_1 Follow in a Divided Interval Probability Problem?

[nickthegreek]

Homework Statement

We have an interval [0,1], which we divide into k equally sized subintervals and generate n observations. Let's 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 please just guide me... don't give me the answers :)

Homework Equations

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 ?

 

[Ray Vickson]

Homework Statement

We have an interval [0,1], which we divide into k equally sized subintervals and generate n observations. Let's 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 please just guide me... don't give me the answers :)

Homework Equations

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 ?

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).

 

[nickthegreek]

Hi Ray, thanks 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 can't 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 don't know some of the terms in english.

 

[Ray Vickson]

Hi Ray, thanks 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 can't 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 don't know some of the terms in english.

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.

 

[nickthegreek]

(1/10)^n?