3
votes
2answers
35 views

Binomial dependent on a Poisson

I have been working on a problem with a binomial rv dependent on a poisson rv and have worked through to this point: $P(X=x) = \sum_{n=x}^{\infty} \dfrac{n!}{x!(n-x)!} p^x(1−p)^{n−x} ...
0
votes
0answers
21 views

Probability in the urn model without replacement.

In an urn with $p$ total marbles, $p_A$ are white and $p-p_A$ are black, we know that the probability of drawing at least $m_A$ white marbles out of a $m$ without replacement follows the cumulative ...
1
vote
2answers
73 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?
0
votes
1answer
27 views

Binomial distribution or probability intersection

I flip a biased coin, p = 0.5 for getting heads. What is the probability of getting heads 8 times ? Firstly I used probability intersection $$ P(A \cap B \cap C \cap D \cap E \cap F \cap G \cap H) = ...
0
votes
1answer
23 views

Finding $V(X)$ when you don't have a density/distribution function.

I just did the first part of this problem: You have a lot of $50$ items and are taking a sample size of $15$. In the lot $3$ items are defective. The lot is accepted if the number of defective items, ...
1
vote
6answers
53 views

Distribution of a binomial variable squared

If I know $X$ is a binomial random variable, how can I find the distribution of $X$ squared (I know that $P(Y=y=x^2) = p(X=x)$ but does this distribution have a standard name)? In particular, how can ...
0
votes
1answer
44 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
1
vote
1answer
39 views

Binomial Coefficient: monotonically decreasing in this range?

relating to this question, I'd like to ask a further one. Again we have $$f(x)={k-1 \choose x-1} p^x (1-p)^{k-x}$$ We know that this term is maximal for $x=kp$, before increasing, afterwards ...
0
votes
1answer
23 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
0
votes
2answers
26 views

Probability of special configuration of ones in a binary string

Consider the sequence $(X_i)_{1 \leq i \leq L}$ of i.i.d. random variables, where $X_1 \in \{0,1\}$ and $P(X_1 =1) = p$. For a $k \in \mathbb{N}$ define the event $A_{k,L}$ as "all ones in the ...
1
vote
2answers
51 views
1
vote
1answer
41 views

How do you calculate a binomial distribution with k > R as opposed to k = R

I'm given the formula: $\displaystyle P(X = k; n, p) = \binom {n}{k} * p^k * q^{n-k}$ And we need to work out the binomial coefficient by hand, instead of using C(n,r). So I have a question: "Some ...
1
vote
1answer
69 views

Estimation solving for binomial k?

Hello all trying to do an estimation problem at work and wondering if I'm on the right track! I'm running a study and its on the internet. I'm trying to determine how many people I need to show an ...
1
vote
1answer
92 views

Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
0
votes
0answers
33 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
0
votes
1answer
25 views

Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
2
votes
1answer
43 views

Probability of two independent random variables being equal

Assume that $X$ and $Y$ are two independent random variables that follow the binomial distribution of parameters $p$ (the probability of one success) and $n$ (the number of trials). I was wondering ...
1
vote
1answer
106 views

Binomial Distribution - independence

I have the following problem that I'm stuck on a few parts. ...
0
votes
0answers
121 views

Solving sample size of hypergeometric distribution given a specific probability

I am trying to figure out how to calculate the sample size of a hypergeometric distribution, given a population, population successes, and probability. Here is the initial formula: ...
1
vote
1answer
40 views

In a lottery of $90$ numbers a man adds extra $1,2,3$

Consider a lottery where $5$ balls are chosen randomly among $90$ balls numbered from $1$ to $90$. A man cheats adding to the $90$ balls, before the draw, three more balls numbered $1,2,3$. We say ...
1
vote
1answer
54 views

Gambler's ruin, probability of loss, infinite turns

A gambler starts with $\$1$ and bets $\$1$ every turn of a game, where he has the probability $p$ to obtain $\$2$ and $1-p$ to obtain nothing. If $p<1/2$, what is the probability he will eventually ...
0
votes
1answer
30 views

Binomials for getting probability of standard deviation

I have the following problem which I am stuck on the second part. Suppose that $30\%$ of all students who have to buy a text for a particular course want a new copy whereas the other $70\%$ want a ...
0
votes
1answer
36 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
2
votes
3answers
86 views

Probability of 5 cards drawn from shuffled deck

Five cards are drawn from a shuffled deck with $52$ cards. Find the probability that a) four cards are aces b) four cards are aces and the other is a king c) three cards are tens and ...
2
votes
1answer
59 views

Question about balls in urns

Suppose there are $n$ balls in an urn, and $r$ of them are red. I select $m$ balls from this urn at random. What is the probability that at least $k$ of them are red? $m$ must be less than $n$, but ...
1
vote
2answers
124 views

DICE - Rolling at least *k* on *n* six-sided dice - with a twist!

I am putting together a table of dice probabilities for a project I am working on and have found myself intimidated by a little "special case" I'm trying to work with. For determining the probability ...
1
vote
1answer
41 views

Lower bound functional binomial r.v.

I am trying to find a bound of the type $\mathbb{E}(|B-\frac{N}{2}|) \geq C \sqrt{N}$ Where $B$ is a binomial variable with parameters $(N,\frac{1}{2})$. The bound doesn't need to be very tight in ...
1
vote
1answer
79 views

Convexity of Binomial Term

I am reading a book on the probabilistic method, and the following claim was made: $\dbinom{y}{n}$ is convex. Why is this the case?
1
vote
2answers
78 views

Probability problem with binomial/multinomial distribution

Mary knows the answers to $20$ of the $25$ multiple choice questions on the Psychology $101$ exam, but she has skipped several of the lectures, she must take random guesses for the other five. ...
0
votes
2answers
63 views

Pascal's Triangle Proof

Trying to determine a formula for the sum of the entries of the $n$th row of Pascal’s triangle, for any natural number $n$. Any proof will do as I have to determine $3$ different proofs. - So far, ...
0
votes
2answers
77 views

How Do You Calculate Probabilities of Random Events Occuring in Sequence?

So I have a series: $f(x_{n+1})=x_n \pm t$ and $f(x_0)=W$ What I'd like to calculate is the probability in terms of $t$ and $W$ (assuming they're any constant $W>t$) that any $f(x_q)=0$ for all ...
9
votes
2answers
181 views

An extrasensory perception strategy :-)

Inspired by classical Joseph Banks Rhine experiments demonstrating an extrasensory perception (see, for instance, the beginning of the respective chapter of Jeffrey Mishlove book “The Roots of ...
2
votes
1answer
131 views

Bins and balls model - filling first bins [close]

We have $n$ bins and $m$ balls. I want to compute the probability that in the first $k$ bins, $q$ of them will be non-empty. I can throw $m$ balls into $n$ bins in $n^m$ ways. Using Stirling ...
0
votes
3answers
113 views

Probability of choosing coins from a bag (why doesn't binomial coefficient work?)

Studying for my exam and would appreciate some help. I have a bag with 2 pennies, 1 nickel and 1 dime. I pick 2 at random. The solutions say: Pr(PP) = $\frac{2}{4} \times \frac{1}{3}$ = ...
1
vote
1answer
61 views

Simple Random Walk

How to find: $\lim\limits_{N \to \infty}\sum\limits_{m=0}^Nu_m$ where $u_m$=${2m \choose m}p^mq^m$ I know there are two cases to consider depending if $p$ and $q$ are equal or not. I should probably ...
0
votes
2answers
68 views

generating function and binomial distribution - counting

I am trying to understand generating function. I have the following problem: There are 50 students in the International Mathematical Olympiad (IMO) training programme. 6 of them are to be selected to ...
0
votes
2answers
30 views

negative binomial distribution problem

Find the probability that you find 2 defective tires before 4 good ones. There is a chance of a tire being defective at a rate of 5%. From my understanding with the negative binomial distribution we ...
0
votes
0answers
29 views

Analytical solution for binomial equation

Suppose that the random variable $X \sim \operatorname{Binomial}_{n,p}$, and suppose we have $p' \in [0,1]$. I have been asked to solve for the least $n$ such that $P(X \leq 2) = p'$. It was ...
10
votes
3answers
582 views

Prove the lecturer is a liar…

I was given this puzzle: At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are ...
1
vote
1answer
56 views

Binomial distribution . Heads and Tails

Consider a coin with P(Heads) = 2/ 3 . We toss this coin 100 times (assume that the tosses are independent). Determine the probability that we get exactly 45 tails out of the 100 tosses. First, ...
2
votes
0answers
46 views

Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$. Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I ...
1
vote
1answer
653 views

binomial distribution(overbooking plane tickets)

I am having trouble with binomial distribution and this problem: an airplane has 200 seats, but 202 tickets are sold. Assume passengers do not show up with a probability of .03 independently. What is ...
2
votes
2answers
47 views

Binomial distribution false reasoning

While reading the answer of a previous question Binomial Distribution Question (Exactly/At Least $x$ Trials for Success), it got me thinking a little. I know the reasoning must be flawed somewhere, ...
1
vote
1answer
75 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 ...
4
votes
1answer
89 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) = ...
2
votes
0answers
156 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 ...
5
votes
3answers
518 views

How many ways to choose $k$ out of $n$ numbers with exactly/at least $m$ consecutive numbers?

How many ways to choose $k$ out of $n$ numbers is a standard problem in undergraduate probability theory that has the binomial coefficient as its solution. An example would be lottery games were you ...
1
vote
1answer
65 views

Binomial Approximation. Calculate $n$

I have the next question: If it is known that $P(X \geq300)=0.3$ and $X \sim \mathop{Bin}(n,0.2)$ How can one estimate $n$?
1
vote
0answers
83 views

Feller vol. I, probability of 'r' successes in binomial process

This question is about the calculation of probability of at least 'r' successes in a binomial process given in the page 151 of feller:intro to probability:vol-1. Before deriving equation 3.5, text ...