Questions tagged [binomial-distribution]
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. (Def: http://en.m.wikipedia.org/wiki/Binomial_distribution)
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Probability of failing an exam set based on individual exam failure probability
So the question is as follows:
X need to pass four out of five separate tests for certification. Assume that the tests are equally difficult, and that the performance on separate tests are ...
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Showing sufficient statistics
In attempting the following questions I am stuck and feel that both parts are the same...Anyone give hints on how I can proceed?
(a) Suppose that $X$ has a binomial distribution with parameters $(n,...
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Probability of getting $n$ heads from multiple coins each tried several times
Assume we have 10 coins, and each has its own probability of getting a head. We toss each coin a different number of times, say 10 times for the first coin, 15 times for the second coin, and so on. ...
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Variance of Binomial Distribution - formula
This might be a really stupid question... but still here we go:
For the Variance we have two formulas:
(i)$$
\sigma^2:=\sum_{k=0}^n (k-\mu)^2p = \sum_{k=0}^n (k^2-2npk+n^2p^2)p= (\sum_{k=0}^n k^2- \...
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Statistics Possion and Binomial Joined Question?
On average, a litter of cats has 5 kittens. Let X be the number of kittens in a litter of cats.
In 20 litters of kittens, what is the probability that at most 15 litters have exactly 5 kittens?
^^I ...
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second moment distribution of random walk
Considering P(X,N) be the probability of finding a 1 dimensional random walk after N steps at position X. What is the probability distribution of $$<X_N^2>$$ where $X_N$ is the position of the ...
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Show $\lim \left| \left( 1-(1-s)\frac{\lambda_n}{n}\right)^n-\left(1-(1-s)\frac{\lambda}{n}\right)^n\right|\le\lim|1-s ||\lambda_n-\lambda |$
As application of convergence theorem in our probability lecture we want to show the generating function of sequence of binomially distributed random variables converges to the generating function of ...
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Poisson-Approximation of Poisson-Binomial distribution
I'm looking for a proof as to why the Poisson-Binomial distribution can be approximated by the Poisson distribution.
A Poisson-Binomial distribution is a sum of independent Bernoulli trials that are ...
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Binomial Probability Formula (inclusive?)
Having some trouble with these two...
Suppose a health insurance company can resolve 60% of claims using a computerised system, the remaining needing work by humans. On a particular day, 10 claims ...
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Prove $\liminf \left(\frac{n!}{(n-k)! n^k}\right) = 1,\ k =$ constant.
I had this as an assumption in my textbook (in the section Poisson approximation to Binomial distribution) but couldn't prove it.
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Interpretation of Binomial R.V. Proposition
If $X$ is a binomial random variable with parameters $(n, p)$, where
$0 < p < 1$, then as $k$ goes from $0$ to $n$, $P\{X = k\}$ first increases monotonically and then decreases ...
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Binomial to Poisson Approximation : Why does p have to be small
I understandd that as n tends towards to infinity for a Binomial distrobution, it becomes a Poisson distobution and i have completed the proof for this.
However, I am not sure why when approximating,...
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Bounding spectral norm of matrix of binomial entries with small probabilities
Consider an $n \times n$ matrix $X$ where entries
$$
X_{ij} = \begin{cases}
C, & \text{w.p. } p\\
0, & \text{w.p. } 1-p,\\
\end{cases}
$$
where $p$ is very small. ...
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How many tuples of {$a, b, c, \ldots$} satisfy $a+b+c+\ldots \leq n$?
Let $n$ be a non-negative integer and $k$ be a positive integer.
Let $a, b, c, \ldots$ be $k$ non-negative integers such that $a+b+c+\ldots \leq n$.
How many tuples of {$a, b, c, \ldots$} satisfy ...
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Asymptotic Expansion for $\frac1{2^n}\sum_{k=1}^n\frac1{\sqrt{k}}\binom{n}{k}$
Prior Art
The fact that
$$
\lim_{n\to\infty}\frac1{2^n}\sum_{k=1}^n\frac1{\sqrt{k}}\binom{n}{k}=0\tag1
$$
is the topic of this question. An argument using a bit of probability theory gives a first ...
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Probability of the number of heads given a number of coin flips
I've been learning statistics through the website Khan Academy and they have this problem:
Let the random variable X = the # of heads from flipping a coin 5
times. The total number of outcomes ...
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Is a Beta distribution a continuous version of the Binomial Theorem?
The visual appearance of the PDF for a Beta distribution resembles that for the terms in the Binomial Theorem. Is the former a continuous variant of the discrete terms of the latter? Are they ...
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Grade 12: Data Management; Probability/binomial distributions -Homework help needed
Struggling to figure these out, I have tried on my own a few times and have yet to get an answer that is remotely close to the correct one or what would be correct. In answer please include all steps
...
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Finding probabilities in a binomial distribuion
$100$ people are each given a lottery ticket and each of those tickets have the probability $p$ of winning a prize (independent of others)
To find the probability that exactly $10$ people win a prize,...
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Suppose X is Binomial(3,1/2) and Y is Binomial(2,1/2). Let Z = 2X - 3Y. Assume X and Y are independent.
Suppose X is Binomial(3,1/2) and Y is Binomial(2,1/2). Let Z = 2X - 3Y. Assume X and Y are independent.
What is the smallest value of Z?
What is the largest value of Z?
I am unsure on how you would ...
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Difference between a binomial experiment and a Bernoulli Trial?
I'm studying in an introductory statistics textbook and it's confusing me when it mentioned this:
"Any experiment that has characteristics two and three and where n= 1
is called a Bernoulli Trial(...
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Intuition behind why a binomial experiment has to have a fixed number trials
I'm studying an introductory statistics textbook and it mentioned this:
"There are three characteristics of a binomial experiment:
There are a fixed number of trials. Think of trials as repetitions ...
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How is $\binom{2n}{n} \sum_{k=0}^{n}\binom{n}{k} \binom{n}{n-k}=(\binom{2n}{n})^{2}$ [duplicate]
I have been trying to make out how:
$\binom{2n}{n} \sum_{k=0}^{n}\binom{n}{k} \binom{n}{n-k}=\binom{2n}{n}^{2}$ so in essence, showing that
$\sum_{k=0}^{n}\binom{n}{k} \binom{n}{n-k}=\binom{2n}{n}$
...
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Product of binomial coefficients and interesting properties
I recently encounter the following quantity
\begin{eqnarray}
\frac{n^+!n^-!}{n!}\frac{k!}{k^+!k^-!}\frac{l!}{l^+!l^-!}
\end{eqnarray}
$n^\pm,n,k^\pm,k,l^\pm,l$ are all non-negative integers. There ...
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1st Yr Statistics Question: Create an approximate $\alpha$ level test of $H_0 : p_1 = p_2$
Let $X_1$ and $X_2$ be binomial random variables with respective parameters $n_1, p_1$ and $n_2, p_2$. Show that when $n_1$ and $n_2$ are large, an approximate level $\alpha$ test of $H_0 : p_1 = p_2$ ...
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Calculating Probability of an event and its expected value over a large number of independent events
So I am in a casino betting on Black in an American Roulette wheel.
Scenario1: I bet for successive 1000 bets starting with a bet of 1 dollar.
If I loose, I bet the double of amount that I bet ...
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Probability of winning the lottery twice
I'm struggling to get the answer to this question. I believe I have almost solved it but am having trouble getting the final answer.
What is the probability of someone winning the lottery twice?
It ...
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How many times should the dice be rolled to maximise chance of winning? 0
A friend bets you $100 on a game involving two six-sided dice, one red and one green. You choose the number of times the pair of dice will be rolled. You win if the number of times a red 6 is rolled ...
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Finding the estimator of π(1-π) from a random sample of n Bernoulli trials.
A random sample of n independent Bernoulli trials with success probability π results in R successes.
Derive an unbiased estimator of π (1 − π).
So, from what I understand (correct me if anything I ...
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Uniform distributed success probability for a coin
$n\in \Bbb N$. Let $X_1 \sim \text{Uni}_{(0,1)}$ and $X_2 \sim \text{Bin}_{n, X_1}$ conditional on $X_1$. I want to find the distribution function of the law of $X_1$ given $X_2 = k$, i.e. $\Bbb P (...
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binomial distribution problem prove or disprove
Let $Y_n\sim B(10, 2p) \wedge 0<p<\frac{1}{20}$.
prove or disprove:
$\lim_{n\rightarrow\infty}p(|Y_n-20p|>\epsilon) = 0$
my try:
$p(|Y_n-20p|>ϵ)=p(Y_n-20p>ϵ)+ p(Y_n-20p≤-ϵ)
=p(Y_n&...
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Deriving the mean of the binomial distribution
According to textbooks, the mean of a random variable $X\sim\displaystyle\operatorname{Bin}\left(n,\theta\right)$ that follows a binomial distribution is given by $\mathbb{E}\left[X\right]=\theta$.
...
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Proof that for a binomial distribution, varX = npq
I need to show that the variance of a binomial probability distribution Var(X) = npq. You can see a full proof here. I'm working on the $E\left[ { X }^{ 2 } \right] $ term and followed it all until ...
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Computing the odds that max. $11$ items are late, if $132$ items are delivered with $96.2\%$ success rate per item. Why two methods give two results?
Probability of success (p) = 96.2% (Item delivered on time)
Number of repetitions (n) = 132 (Items that has to be delivered)
Question: What is the probability of max. 11 items being late.
Note: ...
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Simple Random Sample $X_1=x_1,X_2=x_2,...X_n=x_n$ taken with replacement, find maximum likelihood estimator for parameter $p$ of $X\sim Binomial(5,p)$
Question: Given a Simple Random Sample $X_1=x_1,X_2=x_2,...X_n=x_n $ taken with replacement, find the maximum likelihood estimator for each of the parameters of the distributions below.
The parameter ...
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Expectation of the inverse of Binomial distribution
Suppose the random variable $X_i$ are i.i.d and follow $Binomial(N,p)$.
Then, can we compute the following expectation?
$E\big[\frac{1}{a+X}\big]$ where $a>0$.
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Testing the probability of a Bernouilli variable
I have a Bernouilli variable which is $1$ with probability $p$. I need to test the hypothesis $H_0:p<\theta$ vs. $H_1:p>\theta$, where $\theta$ is a given constant. The question is to find $n$ ...
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Find $\sum_{i=10}^n (\sum_{r=10}^i \binom{30}{r}\binom{20}{r-10})$=?
Evaluate:
$$\sum_{i=10}^n \left(\sum_{r=10}^i \binom{30}{r}\binom{20}{r-10}\right)$$
I tried this and my result come:
$$n\binom{30}{10}\binom{20}{1}+ (n-1)\binom{30}{11}\binom{20}{2} + (n-2)\binom{30}...
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why $\sum _{k=0}^{n}Binomial(n-1,p) = 1$? [closed]
I'm studying my teacher's lecture and got stuck at the proof of this equation, how can I prove it?
one way I think is something like this but I don't it will lead to the correct answer or not:
$\sum _{...
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A question about close votes (please read it before migrating it to meta)
Suppose, on close.stackexchange.com there are $m$ active users with sufficient reputation to cast close votes and $n$ open (not closed) questions (the number of open questions never changes - whenever ...
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A and B have equal chances of winning a single game. A wants n games and B wants n+1 games to win the match. find the odds in favour of A
I am stuck with this question can someone please give me a hint, how to go about this one. Is the total number of matches 2n and 2n +1?, I'm confused by this one
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Binomial distribution: Cumulative distribution vs Chernoff bound
I am studying Chernoff bounds lately but I am not clear how this is better than cumulative distribution function for binomial distribution.
For example, let's say that ...
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Determining largest and smallest elements in binomial expansion
Would it be possible determining which elements in $(0.9999+\frac{1}{n})^n$ are the largest and smallest?
I know that $$\left( 0.9999 + \frac1n \right)^n = \sum_{k=0}^{n}\binom{n}{k} 0.9999^k \cdot \...
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Deriving the Poisson from the Binomial
In my notes I have the following explanation:
The probability function of the Poisson random variable is $P_X(k)={\alpha}^k \frac{e^{-\alpha}}{k!}$ A Poisson random variable with parameter $\alpha$.
...
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Identifying, Describing a Binomial-type Distribution
Fix $p \in (0, 1), N, M \in \mathbf{N}_{\geqslant 1}$. It is well-known that if $X \sim \textbf{Bin} (N, p)$ and $Z \sim \textbf{Bin} (M, p)$, then $Y = X + Z$ is marginally distributed as $Y \sim \...
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Binomial distribution with nonlinear function of successes
Is there a closed form expression for the following expression:
$$\sum_{j=1}^{N-1} {N-1\choose j} q^j (1-q)^{N-1-j} \frac{c-jd}{e+jd}$$
where $c$, $e$, and $d$ are some real numbers? I wonder if the ...
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Binomial Random Variable Intuition
I have the following question from a stats course about binomial distribution:
A multiple choice test has 10 questions, each with 5 possible answers, only one of which is correct. A student who did ...
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the moment generating function of the negative binomial distribution
According to the textbook I use, it states that:
$X$~$Neg.bin(x;k,\theta) = {n-1 \choose k-1}\theta^k(1-\theta)^{n-k}$
Which I have no problem. The problem arises when I try to find the moment ...
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factorial moment generating function
I'm trying to get the factorial moment-generating function of a binomial random variable. I know that
$F_X(t) = E[t^x] = \Sigma_xt^xp(x)$
so I get $\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$
...
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Sum of dependent Binomial distributions
In one of my classes, we stated that if $X_i$ are independent Bernoulli random variables with p proportion of success, then the distribution of the sum $\sum X_i$ is Binomial(n,p). I already proved ...
