Math Help Forum - Advanced Probability and Statistics

Rao-Blackwellized estimator of a poisson process

StatRookie — Fri, 20 Nov 2009 16:45:31 GMT

Impulses arrive at a circuit according to a Poisson Process at an average rate of λ per minute. This rate is not observable, but the numbers X1, ..., Xn of impulses that arrived during n successive one-minute periods are observed. It is desired to estimate the probability e−λ that the next one-minute period passes with no impulsess.
An extremely crude estimator of the desired probability is
i.e., it estimates this probability to be 1 if no impulses arrived in the first minute and zero otherwise.
The sum

I am confused on how to show that this would be a suffcient statistic. Also I know that
Is the Rao-Blackwell estimator but I cannot manipulate the conditional distribution, we have not reached that point in our probability theory class(Doh).

I believe it should become the below statement; however, I am unable to figure out how to fill in the details.
Lastly, I don't understand how one would show that Sn is complete, and thus prove that this estimator is the UMVUE. I sincerely appreciate any help in understanding these concepts.

comparison Gaussian-Exponential distribution

akbar — Fri, 20 Nov 2009 13:18:41 GMT

Let be independent identically distributed Gaussian variables with mean zero and variance one. Let be independent identically distributed exponential random variables with mean one. How do you prove that there is 0" title="n>0" style="border: 0px; vertical-align: middle;" /> such that:

0.99" title="\mathbb{P}(max(\eta_1,...,\eta_n)\geq max(\xi_1,...,\xi_n))>0.99" style="border: 0px; vertical-align: middle;" />

I thought a proper application of the Law of Total Probability would do the trick, but things don't seem that simple. Otherwise, some fancy convolution... But I'm probably wrong.

Thanks for your help.

Joint PMFs of Multiple Random Variables - Urgent Help

essedra — Fri, 20 Nov 2009 11:18:46 GMT

It says my answers are incorrect. Can anyone help me please? What did I do wrong..?

On a given day, your golf score takes values from range 100 to 109, with probability 0.1, independently from other days. Determined to improve your score, you decide to play on three different days and declare as your score the minimum X of the scores X1, X2, X3 on the different days.

Calculate the PMF of X.
pX(107)=

pX(107)=comb(3,1)*0.1*0.3*0.3=0.027

Px(k)=P(X>k-1)- P(X>k)
Where P(X>k) = P(X1>k, X2>k, X3>k)=(109-k)^3*/(10^3)
pX(101)=

pX(101)= comb(3,1)*0.1*0.9*0.9=0.243;
By how much has your expected score changed as a result of playing on three days?

If PX(100+i)=0.3*(1-i)^2 then E(X)=102.475
If P(X.1 then E(X)=104.5 Difference=102.475-104.5=-2.025

Sum of Independent Random Variables - Moments

azdang — Fri, 20 Nov 2009 00:33:22 GMT

Let be independent, each with mean 0, and each with finite third moments. Show that:

It gives a hint to use characteristic functions, so here is what I tried doing. I used to represent the sum of the Xi's from 1 to n.

= and

Then, we want to show that .

My first question is if what I have done so far is okay. My second question is, where would I go from here? I can't seem to see it. Thank you!

What did I do wrong?

essedra — Thu, 19 Nov 2009 21:56:39 GMT

A prize is randomly placed in one of 13 boxes, numbered from 1 to 13.You search for the prize by asking yes-no questions. Find the expected number of questions until you are sure about the location of the prize, under each of the following strategies.

1. An enumeration strategy: you ask questions of the form 'is it in box k'.

(1/13)*1+(12/13)*(1/12)*2+(12/13)*(11/12)*(1/11)*3+...+(12/13)*(11/12)*...*(1/2)*12=(1+2+...+12)/13=6 but for this answer, it says incorrect. what's wrong about my calculation?

2. A bisection strategy: you eliminate as close to half of the remaining boxes as possible by asking questions of the form 'is it in a box numbered less than or equal to k?'.3*(3/13)+4*(10/13)=3.7692 which is correct.

Test Randomness of Hermitian Matrix

Mauri — Thu, 19 Nov 2009 17:40:53 GMT

I'm working on an experiment which requires me to find the eigenvalues of a large random Hermitian matrix. I had originally thought I could just eye-ball the degree to which my matix is truely random, by comparing a histogram with a Normal Gaussian distribution function.

Is there a preferred way to do this which yields a number representing the degree of randomness? Or, is there an easier but acceptable way.

Also, should I test the real parts separately from the imaginary values? Or, can I just test the distribution of the sqrt of the square of each element?

As a Newbie, I'm concerned whether or not I'm in the right forum. Forgive me if I'm not.

Thanks, Mauri

Product of Random variables - Density

akbar — Thu, 19 Nov 2009 14:48:03 GMT

A random variable has Gaussian distribution with mean zero and variance one, while a random variable has the distribution with the density:

What is the distribution of assuming that and are independent?

One approach is to use the formula of change of variables (e.g. using Jacobian) for a product of random variables (see for example Grimmett and Stirzaker, p. 109) through the map: which gives:
, use independence and integrate over to obtain the result.
But the formula is somewhat cumbersome as it gives 2 different expressions whether you assign or to (no sign problem with ). I assume you get to the same result although the calculation doesn't seem tractable.

Another (simpler?) approach is to use the formula of total probability and use independence. But again it doesn't seem you can go beyond an integral of the form:

Is this correct? Is it possible to do better?
Thanks for any help.

finding expected value

noob mathematician — Thu, 19 Nov 2009 14:44:06 GMT

Let X_1,....,X_n be iid rv with density f and cumulative distribution function F. Let:

I want to find the expected value of

I can't seems to find the answer since the distribution is unknown to me

Finding the pdf of an exponential distribution?

Intsecxtanx — Thu, 19 Nov 2009 13:44:02 GMT

The lifetime (in years) equals , where X has an exponential distribution with mean .

How do you find the p.d.f. of Y? Tank you for your time. (Bow)

question about a "half normal" distribution? Thanks for checking this out.

Intsecxtanx — Thu, 19 Nov 2009 13:40:11 GMT

Let X be N (0, 1)

(a) How do you find the p.d.f of |X|, a distribution that is often called the half normal?

(b) How do you find the expectation of |X| ?

Thank you for your time. (Happy)

queueing/exponentials

mickeytheidiot — Thu, 19 Nov 2009 12:16:58 GMT

Hi I'm having some problems with combining exponential distributions.

Suppose we have a single server, fifo queue. Arrivals and service times are distributed exponentially - we can then work out info like variance, standard deviation and expected response time quite easily.

However, now say customers can be divided into two types - 1/3 long and 2/3 short service times, with each group having service times exponentially distributed with a different lambda. how do i combine the two groups to get the same information?

Thanks very much for any help

Help Me Pliz

koharudin — Thu, 19 Nov 2009 02:41:46 GMT

Hi Friends...
One of my friends ask me about references (ex : journal, paper, ebook , knowledge :) ) of computing median of poisson distribution.

Any help welcome.
Thanks for your concern b4. :)

Sorry for my poor english.

Joint Density Questions

Janu42 — Thu, 19 Nov 2009 02:21:38 GMT

1) Let X and Y be two continuous random variables defined over the unit square. What does c equal if f x,y(x,y) = c(x^2 + y^2)?

2) Suppose that X and Y have a bivariate uniform density over the unit square:
f x,y(x,y) = c when 0 < x < 1, 0 < y < 1 and
0, everywhere else
a) Find c
b) Find P (0 < X < 1/2, 0 < Y < 1/4)

3) A point is chosen at random from the interior of a circle whose equation is x^2 + y^2 less than or equal to 4. Let the random variables X and Y denote the x and y coordinates of the sampled point. Find f x,y(x,y).

4) For the following pdf, find fx(x) and fy(y)
f X,Y(x,y) = 1/x, 0 < y < x < 1. (All these inequalities or less than or equal to, I just don't know how to do that on here.)

Chi-Square Test

statmajor — Thu, 19 Nov 2009 01:15:20 GMT

Two different teaching procedures were used in two different groups of students. Each group had 100 students of similiar ability. These are the results:

Group A B C D F
I 15 25 32 17 11
II 9 18 29 28 16

Data is independent from two respective multinomial distribution with k = 5. Test at the 5% level the hypothesis that the two teaching methods are equally effective.

So let X be the students in group 1, and Y students in group 2.

So the test statistic would be: .

Since it's a multinomial distribution, I know that the expectation is np_i, but how would I find p_i?

Joint density function

Sarmadi — Wed, 18 Nov 2009 18:49:52 GMT

Hello friends!

Its great to be a part of this forum, and I am glad to make my first post here.
There is a question of the joint probability density function that I am unable to get even the idea how to solve.
Kindly help.

Question:
If x and y are two random variables having joint density function:
f(x,y)= 1/8(6-x-y) 0 < x < 2, 2 < y < 4
0 otherwise

Find:
1) P(x<1 ∩ y<3)
2) P(x+y<3)

Thanks in advance friends. :)