Tagged Questions
5
votes
2answers
57 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
0answers
54 views
Expected value of a Poisson sum of confluent hypergeometric functions times (version 2)
In continuation to my question here , what is the expected value of a Poisson sum of the following confluent hypergeometric function:
$$
\sum_{y=1}^{Y} (1/Y)({}_1F_1(y,1,z))
$$
where y is positive ...
0
votes
1answer
92 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
0answers
27 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 ...
1
vote
0answers
52 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
61 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
$$
...
0
votes
1answer
17 views
Summation sign inside an expected value
Would it be correct to assume $E\left[\sum U_i\right] = nE[U_i]$?
I am trying to show that $E[∑(U_i - E[U])^2] = (n-1)(\text{sample variance)}$.
Thanks!
0
votes
0answers
24 views
Find the biggest sum from sequence of number which within a range
I need help. How do I find the greatest sum from sequence of number within a finite range, for example:
Given sequence {2,5,4,3,6} and the range is 11, so how to find the number within the sequence ...
1
vote
2answers
38 views
How is this sum calculated?
We have $N$ letters to $N$ different people, and $N$ envelopes addressed to those $N$ people. One letter is put in each envelope at random. Find the mean and variance of the number of letters ...
1
vote
0answers
27 views
Sums of random variables, one being strictly positive
I have the following problem. Let $X$ be a random variable and $Y$ a strictly positive random variable. Is it true under no further general assumptions that:
$$\mathbb P( X + Y \leq x) < \mathbb ...
0
votes
1answer
39 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 ...
4
votes
2answers
124 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
70 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
109 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 ...
0
votes
1answer
56 views
Probability Summations
The probability of a light bulb lasting T hours is $\exp{10}$ distributed:
$$f_T(t) = \frac{1}{10}e^{-t/10}$$
And the probability that the light bulb will be used H hours is Po(12) distributed:
...
