3
votes
5answers
92 views

$\sum\limits_{n=1}^\infty n(\frac{1}{2})^{n}$ [duplicate]

I am trying to find the expected value of the number of even numbers rolled before the first odd number when rolling a fair die until an odd number comes up. I arrived at $\sum\limits_{n=1}^\infty ...
1
vote
1answer
49 views

How many random cards picks with replacement are required?

You pick 1 card from a standard deck of 52 cards. Then put it back in, and pick a card again. Then put it back in and pick a card. etc... How many times do you have to repeat in order to have ...
1
vote
2answers
80 views

probabilistically what can we say about the next throw of a coin after n throws

this may sound easy or hard or whatever but i cant seem to find anything after searching around for a similar question/answer The question is this: What can we say (probabilistically) about the next ...
1
vote
3answers
30 views

A question about sum of probability… if $P(X\ge n)-P(X=n)=P(X>n)$

If we know that $P(X\ge n)=(1-p)^{n-2}$ (This is not the main subject of the question, so I wont explain about it, hope this OK, but in sort: we get it because $P(X\ge n)=\sum_{k=n}^\infty ...
0
votes
1answer
30 views

Convergence in Probability for a sequence

Given sample space $\Omega=[0,1]$ and P( ) the uniform probability measure define random variable $X_1,X_2,.....$ by $X_{2n}=\begin{cases} e^{2n} & \text{if $\omega\ \epsilon\ [0,\frac{1}{2n}]$} ...
1
vote
1answer
35 views

What is the name of this sequence/progression?

Does the following sequence form some special sequence/progression (such as arithmetic progression, geometric progression, hypergeometric progression, and more): $$ p_k: = \frac{\lambda^k}{k!} ...
0
votes
1answer
34 views

Discrete Math Subsets of a Set

A pizza parlor has six meat toppings and four vegetable toppings that can be added to a pizza. Pizzas also come in three different sizes. How many pizzas can be ordered that have at least one meat ...
0
votes
1answer
23 views

How to find sequence from set of values

I have set of probability values arranged in ascending order, p1<p2<p3<...<pM. Now I want to assign set of numbers in the same manner in which probabilities are increasing. It means I ...
1
vote
1answer
34 views

Convergence condition of infinite cosine product

Please show that, given that $\sum_{k\ge1}c_k^2=\infty$ and $c_k\rightarrow 0$, $$\lim_{n\rightarrow\infty}\prod_{k=1}^n\cos{tc_k}=0$$ for every $t\neq0$. (All variables here are real numbers.) The ...
1
vote
0answers
62 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
1
vote
1answer
49 views

Is there a sequence of i.i.d. random variables that is eventually monotonically decreasing?

Here is the problem I'm struggling with: Let $(X_n)$ be is a sequence of independent and identically distributed random variables. What is the probability that the sequence is monotonically ...
3
votes
4answers
78 views

Finding generating functions - how was this jump made?

I'm going through examples of probability-generating functions in a book and am confused by the following example: $$1+2s+4s^2+...=\sum_{n=0}^\infty (2s)^n=(1-2s)^{-1}$$ I understand the summation but ...
0
votes
1answer
56 views

Frequency analysis/discrete uniform distribution in multiple choice tests

I may be using the wrong terms in the title but I read that if something is random then each character will occur an equal amounts of times. I read this when reading about the One-Time Pad cipher, ...
2
votes
1answer
50 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
0answers
47 views

Conditions for Martingale

Let $X{}_{1},X_{2},\ldots$be a sequence of independent RV and let $f{}_{j}$be continuous functions. Let $S{}_{0}=1$ and $S{}_{n}=\sum_{j=1}^{n}f_{j}\left(X_{j}\right)$. Find a necessary and sufficient ...
0
votes
1answer
107 views

Expected length of generating a pattern (throwing dice)

A fair die is tossed repeatedly. The experiment ends as soon as the last six outcomes form the pattern 131131 What is the expected length (i.e. the number of rolls of the die) of this experiment?
0
votes
2answers
42 views

What is the statistical difference (if any) between these two methods of generating an n-digit random number?

To preface, this question is coming from a software developer so it's written from that perspective. If I need to generate a random number with $n$ digits, I could do it in one of two ways. a. Ask a ...
1
vote
1answer
111 views

The normal approximation of Poisson distribution

(I've read the related questions here but found no satisfying answer, as I would prefer a rigorous proof for this because this is a homework problem) Prove: If $X_\alpha$ follows the Poisson ...
1
vote
1answer
40 views

Average of sequence random variables

Let $X_1, X_2, X_3, \dots$ be a sequence of random variables that converges almost surely $$(X_n) \rightarrow X$$ to a number $X \in \mathbb R$ (or more precisely the delta dirac distribution ...
0
votes
1answer
25 views

Probability to find $1$ in a random sequence of bits

Given the random sequence of $N$ bit formed by $0$ and $1$ obtained flipping a coin and associating for example $1$ to head and $0$ to tail, suppose, after having built the sequence, we randomically ...
1
vote
0answers
71 views

Probability of a sequence

For all $k = 1, 2, ...$, consider $f_k: \mathbb{R}^k \rightarrow \{1, 2, ..., C\}$, for some given positive integer $C$, with the following properties. For all $k$, if $(y_1, ..., y_k)$ is a ...
2
votes
1answer
101 views

$L^1$ convergence of a sequence of stochastic integrals and convergence of their quadratic variations

On a filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$ containing a Brownian motion $W_t$. Let $\sigma^n_t$ be a sequence of square intergable adapted processes and consider: ...
2
votes
1answer
49 views

Independent sequences

Let $\{x_i\}_{i = 1, ...,n}$ , $\{y_i\}_{i = 1, ...,n}$ be sequences generated by a pseudo-random number generator using different seed keys, for example $ x_0$ and $y_0$. Are $\{x_i\}$ and $\{y_i\}$ ...
1
vote
1answer
73 views

Probabilities for $1$-in-$n$ events over $n$ trials

I know there are lots of related questions on here, but I can't seem to find what I'm looking for. Given some event with, say, a $1$ in $1,000,000$ probability (e.g., $7$ being chosen randomly as a ...
0
votes
1answer
99 views

What is this the name of this idea? (combinatorics)

The problem: There are three screws, each one a different type {Phillips, Robinson, Slotted}. There are three sets of screwdrivers, each set corresponds to a type of screw. There are no two ...
0
votes
2answers
66 views

How many sequences of n letters chosen from { A,B, …, Z } are in non-increasing, or non-decreasing order

I am studying for a test and this is one of the practice questions. I really don't understand how to start this? It looks like a derangement question to me but I might be overthinking it
1
vote
0answers
100 views

Probability of having a path of a given length in a random graph

Suppose $G=\langle V,E \rangle$ is a directed graph consisting of $n\in \mathbb{N}$ vertices. Vertex $v_i \in V$ has an edge to vertex $v_j \in V$ with a probability of $P(i, j) = f(|i-j|)$ where $f$ ...
0
votes
1answer
31 views

Do greater probabilities approach expected average values with a smaller series?

Let's say you flip a coin 12 times with a goal of getting 6 heads. Then you roll a six sided die 12 times with a goal of getting 2 "ones" faces up. My intuition tells me that rolling the dice has a ...
1
vote
1answer
18 views

What value of $k$ makes the function $p(s)=k*(2/3)^s$ in a probability function?

This is for $s=2,3,4,...$ I'm looking at my old notes on probability and my original answer was $k*\frac{1}{1-\frac{2}{3}}=1$ implies $k=1/3$ because the sum of such an infinite geometric series is ...
4
votes
1answer
85 views

Simplify $\sum_{k=0}^n \frac{1}{k!(n-k!)}.$

Is there a way to simplify the expression $$\sum_{k=0}^n \frac{1}{k!(n-k)!}?$$ This came up when I was trying to determine $\mathbb{P}(X+Y =r)$ given a joint mass probability $$m_{X,Y}(j,k) = ...
0
votes
1answer
110 views

Finding moment generating function of a discrete random variable (series)

So the question is find the moment generation function for a random variable X with pmf $f_X(x)={{r+x-1} \choose x} p^r (1-p)^x ,x=0,1,2,\ldots, 0<p<1,\mbox{ and }r\in\mathbb{Z}^+$. So it's a ...
4
votes
1answer
35 views

When does infinite inclusion-exclusion work if $\sum_n P(A_n) < \infty$?

If $\sum_n P(A_n) = \infty$ then obviously we can't try to apply inclusion-exclusion directly to evaluate $P\left(\bigcup_n A_n\right)$, without taking limits. But what if $\sum_n P(A_n) < \infty$? ...
2
votes
0answers
115 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
0
votes
3answers
31 views

distant land Probability Question

In a distant land, parents continue to have children until they have a girl and then they stop having kids. Assume there is no limit to the number of births possible to each couple. (a) What is the ...
2
votes
0answers
95 views

Is there any efficient algoritm to solve the given $2$ equations?

My equations are as follows: $$\alpha(A,B)=\frac{\sum_{n=1}^\infty P_0[T=n|A,B]\left(nK_0-s_n\right)}{\sum_{n=1}^\infty P_1[T=n|A,B]r_n-\sum_{n=1}^\infty P_0[T=n|A,B]s_n}$$ ...
1
vote
2answers
59 views

Convergence of sequence of probabilities (competition problem)

Let $\{p_i\}_{i=1}^\infty$ be a sequence of real number such that $0<p_i<1$ for all $i$. For each $n \in \mathbb{N}$ we assemble a subset $A_n$ of $I_n=\{1, 2,\cdots, n\}$ as follows: For any ...
2
votes
1answer
158 views

3 - Venn Diagram Question

Finding some serious problem with the famous "Venn Diagram" probability problems. I have "worked out" three such problems associated with a diagram and am wondering if you could pick out flaws in my ...
1
vote
2answers
233 views

Convergence implies lim sup = lim inf

Could someone please explain to me how the following can be proven? I get the intution but don't know how to write it rigorously. Thank you.
2
votes
2answers
794 views

Probability that an integer number having Poisson distribution is even

The probability $P(X=n)$ that an event X takes place $n$ times in a fixed period of time follows the Poisson distribution with parameter $\lambda$ i.e. $$ P(X = n) = e^{-\lambda} \frac{\lambda ^ ...
2
votes
1answer
88 views

The limit of $\frac{|A_n|}{n^2}$

Let $A_n=\{(i,j)\in\mathbb{Z}^2:\gcd(i,j)=1, \ \ 0 \leq i,j\leq n \}$. How to prove the existence of $\lim_{n\to\infty}\frac{|A_n|}{n^2}$, and how calculate this limit? Thank you!
0
votes
1answer
80 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 ...
6
votes
4answers
919 views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
1
vote
1answer
86 views

Expected value of a stochastic harmonic series

It doesn't seem straightforward to put this into mathematical notation, but I'll do my best to explain the setup. Consider a harmonic series of the following type. For the sake of argument, say we ...
3
votes
1answer
68 views

distribution of $\cos(\omega_0 n)$ where n are integers?

Assume we have the sequence $\,x[n]=\cos(\omega_0 n),$ where $n$ are integers. If we suppose these are realizations of a random variable, what would be the p.d.f. of that random variable?
1
vote
1answer
105 views

How to reduce this series / sumation?

I was solving a probability problem and I ended up with the solution of the form : $$ ...
2
votes
1answer
126 views

Prove that $\limsup_{n\to\infty} |X_n|/n \le1 $ almost surely

Suppose {$X_n$} a sequence of random variables. If $\sum_{n=1}^{\infty}P(|X_n|>n)<{\infty}$ Prove that $$\limsup_{n\to\infty}\frac{ |X_n|}{n} \le1 $$ almost surely What i have done so far: I ...
0
votes
0answers
66 views

Sum involving hypergeometric function

I think that the following equality holds: $$\alpha=\sum_{x=0}^\infty \frac{x-C}{y-C}\binom{x+y}{x}\frac{(1-\alpha)^{x+y}(1+C)^yC^x}{(1+(1-\alpha)C)^{x+y+1}} ...
0
votes
1answer
43 views

How to generate sequence like this?

Can you tell what algorithm can generate sequence $x_1, x_2, x_3, x_4, ...$ satisfying: $x_n$ is real, and always $0<x_n<1$. Every change between $x_n$ and $x_{n+1}$, such as increase or ...
2
votes
3answers
68 views

Relation in probability [duplicate]

As part of the solution of an exercise I have the following relation: $$\sum_{k=0}^{\infty}k(1-p)^{k-1}=\frac{1}{(1-(1-p))^2}$$ Where $p$ is a probability. I don't understand where this is coming ...
2
votes
4answers
1k views

Mutually Exclusive Events (or not)

Suppose that P(A) = 0.42, P(B) = 0.38 and P(A U B) = 0.70. Are A and B mutually exclusive? Explain your answer. Now from what I gather, mutually exclusive events are those that are not dependent upon ...