0
votes
0answers
41 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
2
votes
2answers
99 views

How to prove from the definition that $X_n \xrightarrow{\mathbb{P}} X$ implies $\frac{1}{X_n} \xrightarrow{\mathbb{P}} \frac{1}{X}$?

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
1
vote
2answers
529 views

Proof that negative binomial distribution is a distribution function?

In my textbook, a clear proof that the Geometric Distribution is a distribution function is given, namely $$\sum_{n=1}^{\infty} \Pr(X=n)=p\sum_{n=1}^{\infty} (1-p)^{n-1} = \frac{p}{1-(1-p))}=1.$$ ...
1
vote
1answer
188 views

Alternative proof or verification of given proof of convergence in probability

I am asked to show that if $X_n \rightarrow c$ in probability and if $g$ is a continuous function, then $g(X_n) \rightarrow g(c)$ in probability for a statistics homework problem in a section titled ...
6
votes
1answer
423 views

Does law of large numbers converge in $L^1$?

I've seen the law of large numbers stated mainly in two (or three) forms: $S_n/n$ converges in probability (weak law) and converges almost surely (strong law). Also, there is convergence in the ...
1
vote
1answer
81 views

$ P\left(E_e\right)=\sum _{\eta =1}^{\mathbb{H}} P\left(H_{\eta }\right) P\left(E_e|H_{\eta }\right) $ How to prove?

Is it that true? If yes, how to prove this? $$ P\left(E_e\right)=\sum _{\eta =1}^{\mathbb{H}} P\left(H_{\eta }\right) P\left(E_e|H_{\eta }\right), $$ where $E_e$ is an generic evidence, ...