1
vote
1answer
24 views
+50

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

We have a set of $t$ independent random variables $X_i \sim Bin(n_i, p_i)$. We know that $$Pr[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is there an easy way to ...
0
votes
1answer
40 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
28 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
23 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
2answers
52 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
36 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
24 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
17 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, ...
1
vote
0answers
45 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
73 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
30 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
58 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
35 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
57 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
484 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
60 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
58 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
53 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
85 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
35 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
74 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
94 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
46 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
50 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
610 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
141 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
228 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
82 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
89 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
161 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
322 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
177 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 ...
6
votes
2answers
227 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
172 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
118 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
80 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 $$ ...
0
votes
1answer
42 views

Summation sign inside an expected value

Would it be correct to assume $E\left[\sum U_i\right] = nE[U_i]$? I am trying to show that $E[∑(U_i - E[U])^2] = (n-1)(\text{sample variance)}$. Thanks!
1
vote
2answers
54 views

How is this sum calculated?

We have $N$ letters to $N$ different people, and $N$ envelopes addressed to those $N$ people. One letter is put in each envelope at random. Find the mean and variance of the number of letters ...
1
vote
0answers
38 views

Sums of random variables, one being strictly positive

I have the following problem. Let $X$ be a random variable and $Y$ a strictly positive random variable. Is it true under no further general assumptions that: $$\mathbb P( X + Y \leq x) < \mathbb ...
0
votes
1answer
264 views

Handling summations with two variables

If I have a summation with let's say $x=0 \dots 500$ and $y=0\dots1500$ $500 \choose x$ $ 1500 \choose y$ $\dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}$, How would I handle the constant? If I ...
7
votes
2answers
731 views

Random sum of random variables

Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?
2
votes
1answer
91 views

Having difficulty with Summation

How would I compute: $$\sum_{n=2}^\infty \frac{1}{n^2 - n} \cdot n$$ Hints or step by step process would be the most helpful.
-2
votes
4answers
1k views

Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial

The problem: X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$ Expected value is $\sum{x_ip(x_i)}$ But ...