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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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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 $e^{a+b}$).
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1 vote

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1 vote

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

• 379k
Accepted

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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Accepted

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Accepted

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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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