Statistics problem-exponential approximation

In summary: The next thing is to write out the factorials in the actual form they'll take on. I don't think I'll do it here, but I'll point out one thing, that for large n, \frac{n}{n+1} = 1 - \frac{1}{n+1} \approx 1 - \frac{1}{n} . You should be able to use that to simplify the formula.For part b), how do I set up an exponential approximation? To get started, I think that it would be e^(-1) + e^(-2) +...+ e^(1-k)... am I on the right track?The formula you wrote is an exponential approximation, but it
  • #1
nuagerose
12
0
Statistics problem---exponential approximation

Homework Statement


A box contains 2n balls of n different colors, with 2 of each color. Balls are picked at random from the box with replacement until two balls of the same color have appeared. Let X be the number of draws made.

a) Find a formula for P(X>k) k=2,3,...

b) Assuming n is large, use an exponential approximation to find a formula for k in terms of n such that P(X>k) is approximately 1/2. Evaluate k for n equal to one million.


Homework Equations





The Attempt at a Solution



For part a), I got that P(X>k) = (2n-2)/2n * (2n-4)/2n *...* (2n-2k+2n)/2n for k terms.

For part b), how do I set up an exponential approximation? To get started, I think that it would be e^(-1) + e^(-2) +...+ e^(1-k)... am I on the right track?
 
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  • #2
nuagerose said:
For part a), I got that P(X>k) = (2n-2)/2n * (2n-4)/2n *...* (2n-2k+2n)/2n for k terms.

I haven't done the counting, but this formula looks wrong since it appears to have ##n-k## terms. Once you work out the right form, I would clean it up by canceling common factors of 2 and also using the factorial function.


For part b), how do I set up an exponential approximation? To get started, I think that it would be e^(-1) + e^(-2) +...+ e^(1-k)... am I on the right track?

They could mean use http://en.wikipedia.org/wiki/Stirling's_approximation]Stirling's[/PLAIN] [Broken] approximation, which is useful whenever you have an expression for the factorial of a large number.
 
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  • #3
nuagerose said:

Homework Statement


A box contains 2n balls of n different colors, with 2 of each color. Balls are picked at random from the box with replacement until two balls of the same color have appeared. Let X be the number of draws made.

a) Find a formula for P(X>k) k=2,3,...

b) Assuming n is large, use an exponential approximation to find a formula for k in terms of n such that P(X>k) is approximately 1/2. Evaluate k for n equal to one million.


Homework Equations





The Attempt at a Solution



For part a), I got that P(X>k) = (2n-2)/2n * (2n-4)/2n *...* (2n-2k+2n)/2n for k terms.

For part b), how do I set up an exponential approximation? To get started, I think that it would be e^(-1) + e^(-2) +...+ e^(1-k)... am I on the right track?

You really ought to explain the logic: your formula is the probability that the first k colours are all different. Of course, you can cancel out all the 2s to get
[tex] P(X > k) = \frac{n-1}{n} \frac{n-2}{n} \cdots \frac{n-k +1}{n}[/tex]
(Note: the final factor is NOT what you wrote, but I assume that was just a 'typo', since you otherwise seemed to know what you were doing.)

You can do something similar to what Feller would do in his probability textbook, and re-write the result as
[tex] P(X > k) = \left(1 - \frac{1}{n}\right)\left( 1 - \frac{2}{n}\right) \cdots
\left( 1 - \frac{k-1}{n}\right) [/tex]
That provides a convenient starting point.
 

What is exponential approximation in statistics?

Exponential approximation is a statistical method used to estimate the values of a variable based on an exponential function. It involves fitting a curve to a set of data points and using that curve to predict the values of the variable at other points.

What are the assumptions made in exponential approximation?

The assumptions made in exponential approximation include a linear relationship between the logarithm of the variable and the predictor variable, constant variance of the residuals, and normally distributed errors.

How is exponential approximation different from linear regression?

Exponential approximation and linear regression are both methods of fitting a curve to a set of data points. However, exponential approximation is used when the data follows an exponential pattern, while linear regression is used when the data follows a linear pattern.

What are the advantages of using exponential approximation?

Exponential approximation allows for the prediction of values for a variable based on an exponential function, which can be useful in cases where the data is non-linear. It also provides a way to analyze the relationship between variables and identify trends over time.

What are the limitations of exponential approximation?

Exponential approximation is based on several assumptions and may not provide accurate results if these assumptions are not met. It also requires a relatively large amount of data and may not be suitable for small or sparse datasets.

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