MHB Probability distribution of a RV is a function of another RV

bincy
Messages
38
Reaction score
0
Dear friends,

I have a Random Variable I. Sample space of I is from 1,2,3... inf(countably infinite). It's probability distribution P(I=i) is a function of another set of Random Variables Xi's, which are uniformly distributed in [0,1]. These Random variable are iid. I have to find out the mean of I. The mean of Xi's is 0.5 for every i.

I am giving the probability distribution of i below.

[math]P(I=i)=\left\{ \prod_{j=1}^{i-1}\left(1-\left(N*x_{j}*\left(1-x_{j}\right)^{N-1}\right)\right)\right\} N*x_{i}\left(1-x_{i}\right)^{N-1} [/math]

The average of I is [math]\sum_{i=1}^{inf}P(I=i)*i [/math]

Can anyone give any idea to solve this problem?regards,
Bincy

---------- Post added at 16:41 ---------- Previous post was at 15:13 ----------

[math] \int_{0}^{1}\int_{0}^{1}.....\int_{0}^{1}\left\{ \prod_{j=1}^{i-1}\left(1-\left(N*x_{j}*\left(1-x_{j}\right)^{N-1}\right)\right)\right\} N*x_{i}\left(1-x_{i}\right)^{N-1} *i [/math]

Is it the actual average? If so, how to simplify it? Here integration is infinite times.regards,
Bincy

---------- Post added at 17:44 ---------- Previous post was at 16:41 ----------

I would like to add one more general point and kindly seeking the confirmation from you.

That is,

If the prob distribution of a RV X is a fn of another RV Y, X itself would be a fn of Y.
 
Mathematics news on Phys.org
The answer(mean of I) is N since the probability of success in the 1st, 2nd, 3rd and so on is geometric with parameter 1/N. Ignore my 2nd and 3rd posts
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top