0
votes
0answers
12 views

Hermite polynomials with non standard variance (not equal to one)

It is known for probabilists that if $p(x)=\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ then the derivatives of $p$ can be expressed in terms of the Hermite polynomials as follows: $$\frac{d^n}{dx^n} ...
0
votes
1answer
32 views

Condinational Probability of a multinomial random variable

Let $\mathbf X=(X_1,X_2,X_3)$ be a multinomial random variable having the probability density function defined by: $$f_{\bf X}({\bf x})=\dfrac{n!}{x_1!x_2!x_3!}p_1^{x_1}p_2^{x_2}p_3^{x_3}$$ with ...
0
votes
0answers
20 views

Hypothesis testing CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
1
vote
1answer
37 views

How to calculate convolution of two logarithmic function, i.e. $\int \ln(\tau)\ln{(x-\tau)}d\tau$

Here I have a problem to calculate the probability density function (PDF) of the sum of two independent random variables (RVs), $Z_0,Z_1$, and the PDF of $Z_0,Z_1$ are as follow: ...
1
vote
1answer
45 views

Proof/derivation of the sum of a sequence

Question along the lines, "Use a sum formula to derive a closed form formula" for a section covering combination theorems $$ \mbox{Integer function:}\quad {\mathrm f}\left(\,n\,\right) =1 + 4 + 7 + ...
3
votes
0answers
33 views

Maximize or find an upper bound of the function $kx^{k-1}\exp(-\mu(x^k-x))$

I was programming some random variable simulation using the acceptance-rejection method and I encounter with the Weibull$(k,\lambda)$ distribution. This random variable is posible to simulate with ...
2
votes
1answer
71 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
1
vote
2answers
113 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
2
votes
1answer
35 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
0
votes
2answers
32 views

Reliability Probability problem

What is the Probability that at least one close path is formed from A to B where each switch has a Probability of close = p and each switch acts independent of the other Proposed Solution Let ...
0
votes
0answers
18 views

Conditional expectation for discrete random variables

Is it correct that for two discrete random variables $X,Y$ we just have $$E(X|Y \in A) = \sum_{x \in ran(X)} xP(X=x|Y \in A)?$$ This should follow from $$E(X|Y \in A) = \sum_{y \in A}E(X|Y = {y}) ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
2
votes
1answer
46 views

Levy processes, vanilla option and Fourier Transform

The context to this problem is mathematical finance, although the answer does not need specific knowledge of the area. I am trying to work out the expression for the price of a call option using Levy ...
0
votes
2answers
39 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
0
votes
0answers
31 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
2
votes
1answer
181 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
1
vote
3answers
54 views

Chebyshev inequality for $n=1$?

Wikipedia suggests that Chebyhev's inequality is only true for $n \ge 2$, but I don't see why we have to exclude the case $n=1$? Is wikipedia right? Chebyshev
0
votes
0answers
313 views

How can I derive the PDF from conditional probabilities?

I have some function $P(i)$ which is the probability of success for an experiment on the $i$th trial. The probability mass function for the first successful trial is: $$PMF(n) = \left( ...
2
votes
0answers
59 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
0
votes
0answers
26 views

Identically distributed and same characteristic function

If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
0
votes
1answer
16 views

Mean of cumulative distribution function

Suppose a CDF given by: $F(x)=0$ if $x < -2$, $0.6$ if $-2 <= x <= 1$, $1$ if $x > 1$. How can I then calculate $E[X]$ and $E[X^2]$? I found that $E[X]=0$ since there are only jumps, so ...
4
votes
2answers
238 views

Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
0
votes
2answers
40 views

Memorylessness and its square

If we have that $T$ is a memoryless random variable, how do we know if $T^2$ is one too? I am supposed to investigate the cases $T: \Omega \rightarrow \mathbb{R_{\ge 0}}$ and $T: \Omega \rightarrow ...
0
votes
1answer
20 views

Different ways to give cards

The question is: You have 48 cards and want to distribute them to 4 players(so everyone gets 12). How many ways are there to do so? My idea was to take 4 cards in each step and distribute them to the ...
0
votes
0answers
37 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
1
vote
1answer
29 views

choosing the function of a random variable with the lowest variance w.r.t.o the mean of that random variable?

Consider a real gaussian random variable with mean $\theta$ and unit variance. Let $y$ be an observation of the random variable. The objective is to estimate $\theta$ over all possible $y$. Let ...
0
votes
1answer
49 views

Independence random variables

I found two theorems in my notes and they seem to be somewhat complementary which made me doubt that both of them are true: a) Let $X,Y: \Omega \rightarrow \mathbb{R}$ be a measurable function and ...
0
votes
1answer
88 views

Distribution of ceiling function and absolute value of random variable

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is ...
1
vote
1answer
35 views

Unbiased estimate $\lambda^2$

Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ? I mean it would sound ...
0
votes
1answer
97 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
0
votes
2answers
404 views

Version 2:Help finding the probability that $Ax^2 + Bx + C$ has real roots?

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $AX^2 + BX + C$ has real roots? I am given a hint that if ...
0
votes
1answer
33 views

Check for Independence

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...
0
votes
1answer
28 views

Which probability in this hypothesis test?

We have a hypothesis A (null hypothesis) such that $p\le 0.6$ and B such that $p>0.6$. Now we want to develop a deterministic test $\phi$ for 20 people that has a safety of 95%. Hence we would be ...
0
votes
0answers
76 views

If $X_1, X_2$ have exponential distributions what distribution does $Y = \sqrt{X_1^2 + X_2^2}$ have?

If $P_{X_1}(x) = P_{X_2}(x) = k \exp(-k x)$ how will $Y = \sqrt{X_1^2 + X_2^2}$ be distributed? $X_1$ and $X_2$ are independent. What I have Done: The distribution for $Y=X_i^2$ must be ...
1
vote
2answers
42 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
1
vote
1answer
115 views

Expected value, I do not get this “wikipedia triviality”

on wikipedia are two definitions of the expected value for some random variable $E(X)=\int t f_X(t) dt$ and $E(X)=\int X dP$. I do not see how they are equivalent. In cases, that $X$ would be ...
3
votes
1answer
98 views

How to find the CDF of the binomial distribution in terms of an integral

This wiki page says that the CDF of the binomial distribution in terms of the beta function can be expressed as $$F(k;n,p)=Pr(X\leq k)=(n-k){{n}\choose{k}}\int_0^{1-p}t^{n-k-1}(1-t)^k {d}t$$ How to ...
0
votes
1answer
25 views

Meaning of my calculation card game

I have made a calculation and now I do not understand what I did there. It is about the following question: Imagine you have n cards of which there are 2 aces, what is the expectation value to get ...
0
votes
1answer
214 views

Derivatives of quantile functions for continuous distributions

Suppose that $F$ is a distribution function that is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$ with density $f$. Let $F^{-1}$ be the associated quantile function and assume ...
0
votes
0answers
88 views

Deriving the product rule in Jaynes' Probability Theory (p. 28)

I am having difficulty with one of the steps in Jaynes' derivation of the product rule in Probability Theory: The Logic of Science (page 28). It seems like it should be simple calculus, but I don't ...
3
votes
0answers
764 views

How to find probability distribution function given the Moment Generating Function

After searching, I found two questions like mine, but didn't see my answer to my question. Finding a probability distribution given the moment generating function Finding probability using ...
0
votes
1answer
35 views

Can we make piecewise PDFs or CDFs into a single CDF?

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & ...
2
votes
1answer
45 views

Confusion between probability density and probability in EM-paper

I'm reading about expectation maximization from Dempster et al. and there is one point in the paper I get confused about probability density and probability. Maybe you can clarify this to me. Here is ...
1
vote
2answers
45 views

Generally true that $\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y)$?

Is it generally true that $$\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y),$$ where $a$ and $y$ in the expression are constants? To give context to the question, I am ...
0
votes
1answer
46 views

Finding Calculus or Probability error in basic continuous probability problem

I am having a lot of difficulty spotting my error in the following probability problem. The joint probability density function of $X$ and $Y$ is given by $f(x,y) = c\left ...
2
votes
2answers
84 views

Expected value of a series of random variables in a markov chain

I have a Markov Chain such that $X_n = \max(X_{n-1}+\xi _n,0)$ where the $\xi_n$ series is independent and identically distributed. I want to show that if $\mathbb E(\xi_n) > 0$ (where $\mathbb ...
0
votes
0answers
25 views

Finding a (tighter) sufficient condition on the standard deviation of a random variable

Let $\tilde{\phi}$ be a non-negative random variable with a mean normalized to $1$, with $F(\phi) := \Pr(\tilde{\phi} \leq \phi)$ denoting its CDF. $F(\phi)$ is assumed to be twice continuousy ...
0
votes
1answer
80 views

Jaynes' derivation of the laws of probability

I have asked about parts of this before, and now that I understood that explanation, a new doubt showed up in the remaining parts of the derivation. We have the functional equations ...
2
votes
1answer
75 views

Cox derivation of the laws of probability

I have read Jaynes' Probability Theory: The Logic of Science a while ago, but mostly skimmed over parts of his derivations that I didn't immediately understand. Now I'm trying to really understand it, ...
1
vote
1answer
50 views

Bounding the integral of the tails of a random variable.

I found an argument like this in a book, but I couldn't understand how we got this bound. Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all $n \geq 1$, $$ \int_{|X_n| ...