# Tagged Questions

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### Probability of a minimum sum of card values when drawing cards from a custom deck

Apologies if this question has been answered -- I tried searching for an answer but wasn't sure what terminology to use. Suppose I have a deck of cards, consisting of $n_1$ ones, $n_2$ twos, etc. ...
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### How $\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!}=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$

$$P(X\geq m)=\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!};m=0,1,...$$ Show that for any $m=1,2,...$ $$P(X\geq m)=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$$ I couldn't derive it also ...
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### Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$ [duplicate]

Prove that $$\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}.$$ This problem is in the chapter about discrete random variables, but I have no idea what to go about substituting. I can't ...
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### factorial moments of hypergeometric distribution

Factorial moment of positive order : $$\mu_k=\mathbb E[X(X-1)\ldots(X-k+1)]$$ $$=\sum_{m=0}^{n}m(m-1)\ldots(m-k+1)\frac{\binom{a}{m}\binom{b}{n-m}}{\binom{a+b}{n}}$$ ...
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### Double Summation Switch

I have a question about switching the order of summation in this probability question. I'll give the probability background, but it's not necessary to understand my question. We have a random ...
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### Calculating the expected value and the standard deviation of the total profit.

Company sells 2 kind of cars, Lamborghinis and Ferraris, and the sales of the cars are independent. From every sold Lamborghini the company gets 10000 dollar profit and for every Ferrari the company ...
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### A small calculation .

How $\sum_{k=0}^n (-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^k$ i got it $\sum_{k=0}^n(-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^{2n-k}$ And is that $\mathbb E[\mathbb E(X)]=\mathbb E(X)$ ?
I found the formula $$\sum_{n_1=1}^{n-1} \sum_{n_2=1}^{n_1-1} \sum_{n_3=1}^{n_2-1} \cdots \sum_{n_m=1}^{n_{m-1}-1} 1 = {n-1 \choose m}$$ But I don't know how to prove it. Should I use double ...