0
votes
0answers
35 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
0
votes
0answers
29 views

Problems with convergence in mean

I hope you can help me with the following problem Let $\{ e_i : i\in \mathbb{Z}\}$ be and independent U.I. sequence of scalar random variables with zero mean. Let $\{ A_j : j \geq 0\}$ be a sequence ...
1
vote
0answers
39 views

Expectation of $\frac{1}{X+1}$ for a geometric random variable

I am confused over $E(\frac{1}{1+X})$ where $X$ is geometric distribution with parameter $p$. The book wants me to prove that $E(\frac{1}{1+X})=log((1-p)^{\frac{p}{p-1}})$ Here's what I did. ...
2
votes
2answers
45 views

formula for infinite sum of a geometric series with increasing term

I'm looking for the Expectation of the discrete random variable X, E[X], with pmf: $$p(x)=(\frac 16)^{x+1}, x=0,1,2,3...$$ so what I tried is as follows... $$E[X]= \sum_{0}^\infty xp(x) =$$ so then ...
2
votes
1answer
55 views

Calculating the Average Number of Games Required to Reach a Theoretical True Elo Skill Rating from a given Initial Elo Rating

The USCF uses the following formula for Elo rating adjustments: $$R'=R_0+K(S-E)$$ $$E=\frac{1}{1+10^{(R_n-R_0)/400}}$$ Where $R'$ is the new rating $R_0$ is the initial rating $K$ is a ...
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 ...
0
votes
3answers
280 views

What is the probability that A will win…

Two players are rolling two dices, if they get 6 Player A wins, if they get 7, player B wins, else they rolling the two dices again... What is the probability that A will win? I'd like to get any ...
1
vote
1answer
79 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
4
votes
1answer
117 views

Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
1
vote
1answer
18 views

Find the probability of selecting an ordered pair from set $S$

Define $S=\left\{ \left( a,b\right) \in \mathbb{N}\times \mathbb{N} \mid a\leq 10,b\leq 10\right\}$ , randomly choose an ordered pair from set $S$. Find the probability that makes $\dfrac ...
1
vote
1answer
36 views

Iterate through n coins flipping these obtaining all possible combinations.

If I have let say n coins all facing the same way. Is there an iterative method for turning these coins, one at a time, until all possible combinations have occurred one and only one time? This is ...
1
vote
1answer
40 views

ALternate solution to a probability problem

Here is the problem: $A$ and $B$ roll a dice taking turns with $A$ starting this process. Whichever one rolls the first $6$, wins. Find the probability of $A$ winning. I know how to solve this ...
0
votes
1answer
27 views

Why does this hold for the mean hitting time?

Let $X$ be a Markov chain and $T_A$ the hitting time. My text uses this in a proof: $$\mathbb E[T_A \ | \ X_0=k ] = \sum_{l\in S} \mathbb P(X_1=l \ | X_0=k\ )(1+\mathbb E[T_A \ | \ X_0=l ])$$ and I ...
2
votes
3answers
40 views

Confusion Over Sum of Geometric Series

On pg. 88 of A First Course in Probability, it says $$ P_i - P_1 = P_1[(q/p) + (q/p)^2 + \cdots + (q/p)^{i-1})] $$ Therefore: $$P_i = \frac{1 - (q/p)^i}{1 - q/p}P_1 $$ The series on the right in ...
1
vote
0answers
56 views

Sum of poisson random variables

Let $N, X_1, \dots , X_n$ be independent random variables. $N \sim P(\lambda) \quad (\text{Poisson distribution})$, while $X_k \sim B(p)$ (Bernoulli) Let us consider the "random" sum $S = X_1 + ...
0
votes
1answer
83 views

Find the sum of a probability of dice roll that is prime.

Consider rolling n fair dice. Let p(n) be the probability that the product of the faces is prime when you roll n dice. For example, when n = 1, one die is rolled and the probability that the result is ...
38
votes
1answer
988 views

Does a randomly chosen series diverge?

Pick a point at random in the interval $[0,1]$, call it $P_1$. Pick another point at random in the interval $[0,P_1]$, call it $P_2$. Pick another point at random in the interval $[0,P2]$, call it ...
3
votes
2answers
94 views

why generating function $A(z) = 1 + z + z^2 + \cdots$ can be denoted as $\frac{1}{1-z}$

It is easy to see that $1 + z + z^2 + \cdots$ is equal to $\frac{1}{1-z}$ when $1 > z > 0$ and for $z >= 1$, they are not equivalent. So I have thought $\frac{1}{1-z}$ is just a short for the ...
3
votes
1answer
23 views

Expectation of discrete random variable $X$ with $\mathbb{P}[X=n]=\frac{C}{n(n+1)(n+2)}$

Consider a discrete random variable taking only positive integers $n >5$ as values with $\mathbb{P}[X=n]=\frac{C}{n(n+1)(n+2)}$. $$E(X)=\sum_{x=5}^{\infty}x \cdot P[X=x]\\=\sum_{x=5}^{\infty}x ...
3
votes
5answers
111 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
52 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
votes
3answers
162 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
35 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
34 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
39 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
44 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
25 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
55 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
67 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
69 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
81 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
86 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
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
0answers
53 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
127 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
48 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
208 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
43 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
26 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
130 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
52 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
79 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
104 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
71 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
2
votes
0answers
118 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
33 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
19 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
97 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
124 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 ...