0
votes
0answers
28 views

Recurrence Relation Partitioning 8 into 4 parts

Where do I begin with writing down all the partitions of this equation? Let $P(r,n)$ denote the number of partitions of r into n parts. Use the recurrence relation $$P(r, n) = \sum_{i=1}^n ...
0
votes
0answers
13 views

Sum over stochastic processes on the same set of categories

I have a stochastic process consisting of multiple (stochastic) steps, for which I want to know if I can substitute (or at least approximate) it by summing over the deterministic and stochastic parts ...
5
votes
3answers
107 views

Use the binomial theorem to show that for any positive integer $n$, $\displaystyle\sum_{i=0}^{n} {n \choose i} = 2^n$.

Can somebody check to see if this is good enough just to show? It's very simple but the question doesn't say prove or anything like that. So the binomial theorem states that ...
0
votes
1answer
28 views

Probability $\sum_{j=n+1}^{2n+1} {M \choose m+1}{M-m-1 \choose j-m-1}/{N \choose j} $

I have a prob. problem: A school has $N$ students in which $M$ students are leader (of each class in school), and $N>M$. There are $2n+1$ balls in the black box including $n+1$ blue balls and $n$ ...
1
vote
1answer
44 views

Random walks with finite chance of escape

In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A ...
1
vote
2answers
82 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
4
votes
2answers
93 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
0
votes
1answer
57 views

Summing dependent random variables with unknown joint cdf

Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known. Is there ...
1
vote
1answer
37 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
0
votes
1answer
25 views

Average sum of seria

I need some help with the next question in probability: In the range {1,2,...,100}, someone picks randomly 15 different numbers, with the same probability for each number. What is the average sum ...
2
votes
3answers
76 views

Expectations and variance with rolling a dice 10 times

Let's say you roll a fair dice 10 times and X is the number of sides that never show up. (i.e. Roll 1 - 10 = 1424145221, X = 2 because 3 and 6 never show up) Values of $N=0,1,2,3,4,5.\\ P(N=6) = 0$ ...
1
vote
1answer
43 views

Calculating derivation of logarithm of summation of products

I am trying to grasp the idea discussed in this paper. In the second section of this paper it calculates the derivative of (1) which results in equation (2). I cannot figure out how the derivative of ...
0
votes
1answer
33 views

Converting failure rates between periods

I'm trying to figure out how to convert an annual failure rate between periods. Assume failures are uniform and independent. I know that the quick, back-of-the-envelope way is simply to divide the ...
0
votes
0answers
9 views

Proving a disequality involving hypergeometric distribution

How would you prove symbolically the following property? $$H(m_a+1,p_a+1,m,p) < H(m_a,p_a,m,p)$$ where $H(m_a,p_a,m,p)$ is the probability of drawing $m_a$ white balls in a series of $m$ ...
0
votes
1answer
18 views

Probability and Production equation translation

I know that the pi is like a summation except multiplication instead of addition and that P(x) means the probability of, but I'm having trouble putting it all together, esp the $w_i$ such that $w_1, ...
2
votes
0answers
48 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
2
votes
1answer
86 views

How to numerically evaluate the CDF of this random variable?

I have a discrete random variable $X=0,1,2,\ldots$ with the following probability mass function: $$P(X=x)=\sum_{t=0}^x\binom{n}{t}p^t(1-p)^{n-t}\binom{m-t}{x-t}q^{x-t}(1-q)^{m-x}\tag{1}$$ where ...
1
vote
2answers
31 views

Randomized Algorithm

I asked this question earlier but I wanted to change the problem. A band has tour sites A, B, and C. They get paid every time they play at each tour site, specifically: ...
0
votes
2answers
65 views

Show that $\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$.

Show that $\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$. I fell into this hole where I keep finding that $\sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i ...
1
vote
0answers
37 views

Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
0
votes
1answer
63 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
3
votes
1answer
488 views

Is this infinite sum always less than zero?(+500pts bounty for the correct answer)

I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum: $$\frac{\partial}{\partial d}\sum_{n=1}^\infty n ...
3
votes
2answers
66 views

Show that $\sum^{n}_{i=1}(\bar{X_n}-X_i)^2=\sum^{n}_{i=1}[(X_i-\mu)^2-(\bar{X_n}-\mu)^2]$

Suppose $X_1,\dots,X_n$ are i.i.d and $E(X_1)=\mu$ and $\text{Var}(X_1)=\sigma^2$. I have to show that $$\sum^{n}_{i=1}(\bar{X_n}-X_i)^2=\sum^{n}_{i=1}[(X_i-\mu)^2-(\bar{X_n}-\mu)^2]$$ What I did was ...
2
votes
1answer
60 views

expected value of a function of two random variables

I am trying to calculate this sum (which is expected value of a function of two independent Poisson random variables): ...
1
vote
1answer
56 views

How can I represent this as a sum?

I am solving probability, and I got the need to know what this sum is: $ \frac{1}{2} \times \frac{1}{3} + \frac{1}{2}\times \frac{2}{3} \times \frac{1}{4} + \frac{1}{2} \times \frac{2}{3} \times ...
0
votes
1answer
92 views

Markov Chain: I don't understand this solution to a conditional probability problem after n state transitions…

What is the probability of being in state 4 after two steps, given that one is in state 5 after 8 steps? Markov Chain is at top of link. The sample solution is part (f). I have no idea what the ...
1
vote
2answers
73 views

Why does $\sum\limits_{i=1}^n \frac{i-1}{n} = \frac{n-1}{2}$?

I am trying to understand the accepted answer to this question: Find: The expected number of urns that are empty And am stuck on the part I mentioned above. I understand that: $\sum\limits_{i=1}^n ...
0
votes
0answers
42 views

Poisson probability inequality

How do I find $S$ such that the inequality $CL \le e^{-n\lambda R}\sum_{k=0}^S \frac{(n \lambda R)^k}{k!}$ holds, where $n$ is a positive integer representing number of units in service, $\lambda$ is ...
1
vote
1answer
82 views

Change of variable in an infinite sum

I'm currently trying to understand a derivation from WolframMathWorlds. I got to step 6 where a change of variable happens. You can see the equation here. I understand everything except how they get ...
0
votes
1answer
95 views

$P(X-np>n\varepsilon)\leq E\{e^{\lambda \cdot (X-np-n\varepsilon) }\}$

For $X \mathtt{\sim} \text{Bin}(n,p), \lambda > 0, \varepsilon > 0$, how do you show the following? $$P(X-np>n\varepsilon)\leq E\{e^{\lambda \cdot (X-np-n\varepsilon) }\}$$ Unless I made some ...
1
vote
1answer
47 views

Canceling factorials and exponentials in sum

I'm trying to understand the following the proof. I want to show that $$E\left[\frac{1}{X+1}\right] = \frac{1}{(n+1)p}(1-(1-p)^{n+1})$$ The proof goes like this: $$ \begin{align} ...
2
votes
1answer
51 views

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 ...
4
votes
4answers
715 views

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 ...
0
votes
0answers
145 views

factorial moments of hypergeometric distribution

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

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 ...
1
vote
1answer
2k views

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 ...
0
votes
1answer
84 views

How is this step in the proof that $E[X]=\sum_{x=1}^\infty P(X\geq x)$ justified?

From Wikipedia, $$\sum_{i=1}^\infty P(X\geq i)=\sum_{i=1}^\infty\sum_{j=i}^\infty P(X=j).$$Interchanging the order of summation, we have,$$\sum_{i=1}^\infty\sum_{j=i}^\infty ...
-1
votes
2answers
90 views

Simple Paths Along Vertices

Let $v$ and $w$ be distinct vertices in $K_n$, $n\geq 2$. Show that the number of simple paths from $v$ to $w$ is $$(n-2)!\sum_{k=0}^{n-2}\frac{1}{k!}.$$ A path with no repeated vertices is called a ...
0
votes
2answers
78 views

What does this series converge to?

What does the following expression converge to? $${\sum_{i = 1}^n{\left(\frac{S-s_i}{S}\right)^S}}$$ Where the sum of the $s_i$'s equals $S$. How do you work out what it converges to?
1
vote
1answer
186 views

Can someone explain the intuition behind this moment generating function identity?

If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$. But why does: $$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= ...
0
votes
2answers
27 views

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 $\mathbb E[\mathbb E(X)]=\mathbb E(X)$ ?
1
vote
1answer
145 views

Binomial coefficients as multiple sums

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 ...
-1
votes
1answer
425 views

Expected value of sum squared is sum of expected value squared?

Consider the following expression: $ X \sim BIN(1, p) $ $ Var(\bar{X})=Var(\frac{\sum_i{X_i}}{n}) = \frac{1}{n^2} Var(\sum_i X_i) = \frac{1}{n^2} \left( \sum_i E(X_i^2) - ( \sum_i E(X_i) )^2 \right) ...
3
votes
2answers
195 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
7
votes
4answers
297 views

Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$

I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$ I know that $$2^n = ...
6
votes
2answers
249 views

Binomial probability with summation

Show that $$\sum_{k=0}^{m} \frac{m!(n-k)!}{n!(m-k)!} = \frac{n+1}{n-m+1}$$ Attempt: It becomes: $$\sum_{k=0}^{m } \frac{\binom{m}{k}}{\binom{n}{k}}$$ Telescoping, pairing, binomial theorem don't ...
0
votes
1answer
178 views

Expected value of a Poisson sum of confluent hypergeometric functions

How to compute the expected value of a Poisson sum of the following confluent hypergeometric function: $$ \sum_{y=1}^{Y} {}_1F_1(y,1,z) $$ where y is positive integer taking values from the Poisson ...
0
votes
1answer
123 views

Skewness of a sum with a positive summand

Let $X$ and $Z$ be two random variables with finite third moment, and let $Z>0$. Is it true that the skewness of $X+Z$ is greater or equal than that of $X$? Such a relation clearly holds for the ...
2
votes
0answers
81 views

Azuma's inequality with high probabilistic bounds

Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any ...
1
vote
1answer
64 views

Is this sum equal to 1?

Is this function $P:\mathbb{N}\mapsto \mathbb{R}$ such that $$ P(i)=\frac{1}{m^n}((m-i+1)^n-(m-i)^n), \quad i\in\mathbb{N} $$ a probability over natural numbers? I was trying to calculate if $$ ...