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1 vote
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Poisson Distribution Derivation of Expected Value

No, your approach is not quite correct. Since you want the expected daily "unnecessary costs" and this is equal to $350$ times the number of on-call staff who are not called in to replace ...
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0 votes

Calculate $ϕ(\lambda):=\mathbb{E}[e^{-\lambda X}]$ for $\lambda>0$ and $\mu=0 $

What you did wrong is that $$ e^{-\lambda x}e^{-x^2/\sigma^2}\neq e^{\lambda/\sigma^2}e^{x^3} $$ does not hold (it seems that you that $e^ae^b=e^{ab}$ instead of $e^{a+b}$).
1 vote

What is the distribution of the product of three random variables?

Well, just apply the definition twice: $$f_{XYW}(z)=\int_{-\infty}^\infty f_{XY}(t)f_W\left(\dfrac zt\right)\dfrac1{|t|}dt=\int_{-\infty}^\infty\int_{-\infty}^\infty f_X(x)f_Y\left(\dfrac tx\right)\...
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-1 votes

Poisson Distribution Derivation of Expected Value

Hint Let $X$ be the numbers of dispatchers that are absent for work. Then, $X\sim \text{Poisson}(3)$. Let $Y$ be the number of additional dispatchers available that are not called. Then, $$\mathbb E[\...
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1 vote

Prove $\int_{0}^{+\infty} \frac{\exp(-t)}{\sqrt{\pi t}} \exp (-\frac{x^2}{4t}) ~\mathrm{d} t = e^{-|x|}$

Here's an approach that uses (mostly) basic calculus. Lemma (Glasser's master theorem): If $f$ is continuous, $\int_{-\infty}^{\infty} f$ exists, and $a\ge 0$ then $$\int_{-\infty}^{\infty} f\left(x-\...
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How do you calculate the expected value of geometric distribution without diffrentiation?

The problem can be viewed in a different perspective to understand more intuitively. Let's see the following definition. "A person tosses a coin, if head comes he stops, else he passes the coin ...
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4 votes
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Upper bound of joint probability of two RVs $X$ and $Y$, $\mathbb P(X<Y_1,..., X<Y_n)$

For an upper bound, the best that you can do is $$ \Pr(X_1 < Y_1 , \dotsc, X_1 < Y_n) \leq \Pr(X_1 < Y_1) = a. $$ To see that this is tight for any $a$, let $X \sim \mathrm{Bernoulli}(1 - a)$ ...
1 vote

Probability Distribution Calculation of Variables

When they say "for $y = 1,2,3...$, note the dots: "$...$" That means $y$ can take on any positive integer value: the sum doesn't stop at the third term. So you want $$1 = \sum_{y=1}^\...
1 vote

Question from P1 exam book - joint continuous gamma distribution

$$ f_{U,V}(u,v) = \frac {{ \begin{vmatrix} 1 & 1 \\ {\frac{y}{(x+y)^2}} & {\frac{-x}{(x+y)^2}} \end{vmatrix}}^{-1} \cdot (uv)^{(\alpha -1)} \cdot \lambda^\alpha \cdot e^{-\lambda uv} \cdot (u-...
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1 vote
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Independence of Minimum and Maximum of a set of random variables

I assume you already computed $F_m$ and $F_M$, as mentioned in your question: for $x \in [0,1]$, we have respectively $F_M(x) = x^n$ and $F_m(x) = 1 - (1-x)^n$. In particular, $\mathbb{P}(M \le \frac{...
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0 votes

Uniform distribution on the triangle $ ∆:=\left \{ (x, y) ∈ R^2|0 < x, y < 1, x + y < 1 \right \}$

Let $Q= X/Y$ and $S=X+Y$ ($Q$ for quotient and $S$ for sum). Personally, for continuous random variables, I find it easier to find the distribution before the density function. For $Q$: Since $X,Y\...
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1 vote
Accepted

Problem with the minimum distribution of a set of iid random variables

It's $n(1-x)^{n-1}$, with no minus sign before, otherwise you would have a negative density. So up to a change of sign you are correct. I do not know what the other function refers to, it does not ...
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2 votes

Maximum of a sequence is $o(\sqrt{n})$

Fix a positive number $L>0$. You can write $$ \max_{i\leq n} w_i^n \leq \max\left(L,\sum_{i=1}^nw_i^n \mathbf{1}(w_i^n >L)\right). $$ The above inequality is true because either 1. $\max_{i\leq ...
3 votes

How many expected flips before my sausage patties are all face up?

You got an answer already with one interpretation which matches your code, but I'd argue an equally valid (and more mathematically nice, albeit perhaps less reasonable) interpretation described as &...
0 votes

Can you please help me with this question using distribution property to solve this question

Minimize $x.$ In the first question $x$ can be as small as $1.$ How many $yz$ pairs are there if $x = 1$? y can be as small as $1$ and as large as $8.$ So, 8 pairs. And if $x=2?$ 7 pairs. etc. $8+7+...
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3 votes
Accepted

How many expected flips before my sausage patties are all face up?

We use the random patty selection implied by the code, where first a nonzero number and then a set of indices of patties to flip are chosen uniformly at random. Let $E(n)$ denote the expected number ...
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1 vote

Probability of sampling linearly independent vectors

If I am not mistaken, when $q$ is composite without squares, i.e., $q = \prod_{i = 1}^\kappa p_i$ for distinct primes $p_i$, you can use the Chinese Remainder Theorem to establish the probability that ...
2 votes

Sum of i.i.d. random variables for which Chebyshev inequalities are tight

You can replace "infinite" by "2 or more." That is, there is no example where the inequality is tight for two distinct values. With no loss in generality (just change the units ...
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4 votes

Sum of i.i.d. random variables for which Chebyshev inequalities are tight

For your first question, consider a r.v. $X$ with finite first and second moment, variance $\sigma^2>0$ and w.l.o.g. $\mu=E(X)=0$. Assume that the Chebyshev inequality $$P(|X|\geq\varepsilon)\leq\...
1 vote

Difference between Almost sure convergence and Convergence in probability

Almost sure convergence is strictly stronger than convergence in probability. Perhaps an example of a sequence converging in probability but not almost surely might help you? See the first answer of ...
1 vote
Accepted

How to find the distribution of $X$ with $P[X = k \mid X + Y = n] = \binom{n}{k} 2^{−n}$?

Denote $p_j=\mathbb P(X=j)=\mathbb P(Y=j)$. Rewriting the conditional probability gives for $0\leqslant k\leqslant n$, $$ \mathbb P(X=k,X+Y=n)=\binom nk2^{-n}\mathbb P(X+Y=n). $$ Then observe that $\...
0 votes

Random arrangement of English Alphabet; Finding probability that no two vowels are next to each other.

(Assuming repetition is allowed) Here's another way to count it. The number of allowed arrangements is given by sum of first row of $$ \begin{bmatrix} 21 & 5 \\ 21 & 0 \end{bmatrix} ^ {26} $$ ...
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3 votes

Random arrangement of English Alphabet; Finding probability that no two vowels are next to each other.

Addendum added to respond to the comment of Jean Marie. I am assuming that there are exactly $(5)$ vowels in the alphabet. For $k \in \{0,1,2,3, \cdots, 13\}$, let $f(k)$ denote the number of ways ...
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1 vote
Accepted

limiting the log-likelihood function for Weibull distribution

For the sake of convenience, let $$\xi = \sum_{i=1}^n \log x_i = \log \prod_{i=1}^n x_i, \quad g(\alpha) = \sum_{i=1}^n x_i^\alpha; \tag{1}$$ then $$\ell(\alpha) = n \log \alpha + (\alpha - 1) \xi - g(...
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1 vote

Question from P1 probability exam book - conditional probability continuous case

Looks like a typographical error. The expression $$\frac{3y^2}{x^3}I_{(0,x)}(y),\tag{$\ast$}$$ viewed as a function of $x$ with $y$ fixed, does not integrate to $1$ (the integral would run from $x=y$ ...
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0 votes

limiting the log-likelihood function for Weibull distribution

I believe that the issue is about the sufficiency of the solution to the first order condition with respect to the global optimization problem. As you showed that the log-likelihood function is ...
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2 votes
Accepted

Almost Surely Convergence of a Series of Random Variables

Your argument for the computation of the expectation is fine. However, for the convergence of the series, your reasoning does not work. Having $\mathbb E[Y_n]\to 0$ does not mean that $\sum_{n\...
0 votes

$X$~UNI[0,1], $F$ a distribution function, then $Y:= F^{-1}(X)$ has distribution $F$. Looking for intuition in this technical result

I think that I may have asked this question in the past as well, but I think I have the answer now. A simple re-writing makes the lemma much clearer. Suppose a function $F$ satisfies the requirements ...
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An urn initially contains one red and one blue ball question

(a) Notice that $P(X>k)$ is the probability of you geting a red ball from the first $k$ selections, that is $\dfrac{1}{2}\cdot\dfrac{2}{3}\cdot \dfrac{3}{4}\dots \dfrac{k}{k+1} = \dfrac{1}{k+1}$, ...
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1 vote
Accepted

CDF from a probability function

Just extending a little bit the suggestion from @Rik93, using your notation and indicator functions: $$F(x)=\mathbf{1}_{(m-5\le x \le m)}(x)F_2(x)+\mathbf{1}_{(m< x \le m+5)}(x)[F_2(m)+F_3(x)]+\...
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1 vote

Is there a relationship between the mean of a Poisson distribution and the first "impossible" value to come in such distribution?

The answer of course depends on the meaning of "really unlikely." Generally, suppose $N > 0$ and we want to estimate the first $k \ge \lambda$ such that $\mathbb{P}(X = k) = e^{-\lambda} \...
0 votes
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How to go from Radon–Nikodym derivative to classical derivative in change of variables formula of p.d.f.?

We proved in this thread that $\mu_Y$ is indeed absolutely continuous (a.c.) w.r.t. $\lambda$. Lemma: If a finite Borel measure $\mu$ on $\mathbb R$ is a.c. w.r.t. the Lebesgue measure $\lambda$ then ...
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2 votes
Accepted

The absolute continuity of push-forward measure

I formulate @Masacroso's idea as follows. First, we need 3 lemmas. Lemma 1: If $g:\mathbb R \to \mathbb R$ is monotone and differentiable, then $g$ is absolutely continuous (a.c.). Lemma 2: Let $F:\...
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0 votes

1D analytical solution for entropy-regularized Wasserstein distance

The best you can get in the general case is to write down a solution in terms of Lagrange multipliers- which will be functions, not scalars, in the continuous setting. The problem is equivalent to ...
1 vote
Accepted

Degenerate distribution on a sample space of n elements

They are speaking only in the context of this sample space where there are $n$ elementary outcomes, i.e. every distribution can be represented by a tuple $(p_1, p_2, \ldots, p_n)$ where $\sum_{i=1}^n ...
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3 votes
Accepted

If I receive $20$ emails each hour, will it mean that the probability of receiving $10$ emails in $2$ hours will be $\frac{e^{-40} 40^{10}}{10!}$?

Let $X(t)$ be the random number of emails received per $t$ hours. If the average rate of emails is $\lambda = 20$ per hour, then $$X(t) \sim \operatorname{Poisson}(20t),$$ and $$\Pr[X(t) = x] = e^{-...
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0 votes
Accepted

light bulbs distribuited as uniform

Your answer is correct, but it is much, much simpler to reason as follows. Let $A$ represent the event that the highest observed lifetime among $10$ light bulbs is greater than $700$ hours. So we ...
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Do individual Metropolis-Hastings map preserve the target measure?

Happy to be corrected here but the maps are just translations: either you don't move or you move by $\epsilon$ so are definitely measure preserving. The densities of $\eta$ and $\epsilon$ are given so ...
1 vote

mean independence of a sequence of random variables

No, this does not remain true if we change the order. Let $(Y_n)$ be i.i.d. with $\mathbb{P}(Y_n=2)=\mathbb{P}(Y_n=0)=\frac 12$, and define a martingale $(M_n)$ by $M_0 = 1$, $M_n = \prod_{i=1}^n Y_i$....
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3 votes

Difference between the number of heads and tails when tossing a coin

By symmetry, the average (signed) difference between the number of heads and number of tails must be 0. After all, if you simply interchange heads $\leftrightarrow$ tails, the mathematics is just the ...
1 vote
Accepted

Differential entropy for joint distribution, bounded from below by the maximum of the marginals?

No. Trivial counterxample: let $X_1,X_2$ be iid uniform on $[0,\frac12]$ Then $h(X_1,X_2) = -2 $ bits and $h(X_1)= -1$ bits In general: it's still true (chain rule) that: $$h(X_1,X_2) = h(X_1) + h(X_2|...
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1 vote

mean independence of a sequence of random variables

No. For a simple counter-example consider $\{X^{2},X,0,0,\cdots\}$ where $X \sim N(0,1)$.
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3 votes
Accepted

How can I compute the following conditional expectation?

There is a nice way to compute this. We have $$\mathbb E[X+Y|X+Y]=X+Y$$ because $X+Y$ is $\sigma(X+Y)$-measurable. But by linearity and identical distribution $$\mathbb E[X+Y|X+Y]=\mathbb E[X|X+Y]+\...
2 votes

Stochastic process with Student's t-distribution

It's not far off from the Hyperbolic diffusion: $$dX_t = \alpha \frac{X_t}{\sqrt{1+X_t^2}}dt + \sigma dW_t$$ [ cf. Parameter Estimation in Stochastic Diffential Equations, page 2 or example 2.6 on ...
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4 votes
Accepted

Stochastic process with Student's t-distribution

Leonenko and Šuvak call it Student diffusion process. Not too "common", but there's no other name I am aware of.
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0 votes

Confidence two biased dice are the same?

I am thinking of two ways to approach the question. You could go down the path of considering rolling both at the same time and counting when the values match. I believe your PDF would be standard, ...
0 votes
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Calculating $\operatorname{cov}(\lambda\sinh(\frac{z-\gamma}{\delta})+\xi, \sigma z+\mu)$

Let $z=\delta w$ where $\delta>0$ and $\alpha=\gamma/\delta$ so $w\sim{\sf N}(0,1/\delta^2)$ and \begin{align}\Bbb E\left[z\sinh\frac{z-\gamma}\delta\right]&=\delta\Bbb E[w\sinh(w-\alpha)]\\&...
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3 votes

A question about the theorem of characteristic function

This follows from independence and the Fubini-Tonelli theorem. To see this, let $\alpha \in \mathbb{R}$ and $A = \{(u,v) \in \mathbb{R}\times\mathbb{R} : u + \sigma v \leq \alpha \}$. Then, \begin{...
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2 votes
Accepted

Looking for an efficient way of finding the Covariance of Y/X and XY given a shared pdf

Note that $X\sim\text{Gamma}(k=1,\theta=\frac{1}{2})$ and $Y\vert X \sim \text{Uniform}(0,X)$, so $E[X] = \frac{1}{2}$, $E[X^2] = \frac{1}{2}$, $E[Y|X] = \frac{1}{2}X$, and $E[Y^2\vert X] = \frac{1}{3}...
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Looking for an efficient way of finding the Covariance of Y/X and XY given a shared pdf

$$\operatorname{Cov}[U,V] = \operatorname{E}[UV] - \operatorname{E}[U]\operatorname{E}[V] = \operatorname{E}[Y^2] - \operatorname{E}[Y/X]\operatorname{E}[XY]. \tag{1}$$ Then for $a, b \in \mathbb Z$, ...
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