Binomial Definition and 640 Threads

1/(1+x)2 and (1+x)-2 are the same but the negative index binomial has an infinite expansion, but the other as a denominator, stops at x 2 So please help me understand this Thank you in advance.

My approach, ##(0.90)^{2.2}=(1-0.1)^{2.2}-1+\dfrac{2.2×-0.1}{1!}+\dfrac{1.2 ×2.2×(-0.1)^2}{2!}+\dfrac{0.2×1.2×2.2×(-0.1)^3}{3!}+...## ## =1-0.22+0.0132-0.000088=0.7931## There may be other approach. Insight welcome.

Hello everyone, I've been trying to understand the proof for the binomial theorem and have been using this inductive proof for understanding. So far the proof seems consistent everywhere it's explicit with the pattern it states, but I've started wondering if I actually fully grock it because I...

If we had a multiple choice exam with , say, 20 questions, with 4 choices for each question, we can analyze it as a Binomial(20, .25). What if instead , some of the questions offered 2,3, 4, etc., choices? Is there a " nice" way of analyzing the exam as a whole?

##(1+x)^n=1+C_1x+C_2x^2+C_3x^3...+C_nx^n## Let ##x=1##, hence ##2^n=1+C_1+C_2+C_3...+C_n## which is equal to the sum of the coefficients. I originally thought the sum of the coefficients would be ##2^n-1## since the very first term ##1## is just a number and has no variable. But apparently...

Show that for all positive integers ##n##, $$\binom{n}{1} - \frac{1}{2}\binom{n}{2} + \cdots + (-1)^{n-1}\frac{1}{n}\binom{n}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$$

On Wikipedia one can read in the article Beta-binomial distribution: > It also approximates the binomial distribution arbitrarily well for > large ##\alpha## and ##\beta##. where 'It' refers to the beta-binomial distribution. What does 'arbitrarily well' mean here?

This is also pretty easy, ##z^5=(a+bi)^5## ##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}## ##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i## ##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4## ##\bigl(\Im (z))=...

$$(7-6x)^3+(7+6x)^3=1736$$ $$⇒(7^3(1-\frac {6}{7}x)^3+(7^3(1+\frac {6}{7}x)^3=1736$$ $$343[1-\frac {18}{7}x+\frac {216}{98}x^2-\frac{1296}{2058}x^3]+343[1+\frac {18}{7}x+\frac {216}{98}x^2+\frac{1296}{2058}x^3]=1736$$ $$343[2+\frac {432}{98}x^2]=1736$$ $$686+\frac {148,176}{98}x^2=1736$$ $$\frac...

Hello all, I am trying to solve this one: John has n friends . He wants to invite in each evening (365 days a year) three of his friends for dinner. What should be the size of n, such that it will be possible not to invite the same triplet twice ? What I did was: \[\binom{n}{3}\leq 365\]...

What if the value of X is not integer, such as P(X < 1.2)? a) Will the continuity correction be P(X < 1.2 - 0.5) = P(X < 0.7)? or b) Will the continuity correction be P(X < 1.2 - 0.05) = P(X < 1.15)? or c) Something else? Thanks

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate $A.$ Random samples of $200$ voters were polled in each of the...

Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!

[Mentor Note -- Multiple threads merged. @issue -- please do not cross-post your threads] Hi, everyone It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve. Glad to get for help Thanks to all the respondents

Hi, everyone It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve. Glad to get for help Thanks to all the respondents

I know how to expand it. My question is: the expansion has 8 terms so what would be the middle term? Will the answer be "the expansion has no middle term"? Or maybe seeing the phrase of the question (middle terms), there will be two answers (the 4th and 5th term)? Thanks

If we have a spin measurement with P(up)=0.5 en P(down)=0.5, this is equivalent to tossing a coin P(heads)=0.5 and P(tails)=0.5. The probability of having five heads and five tails out of ten tosses is the binomial: ##\binom{10}{5}(0.5)^5(0.5)^5##. So the same would hold for the spin...

So I think I may be overcomplicating this problem but I realize that in order to find the x^3 term it will be the product of the two binomials, ie. x^1*x^2=x^3. The coefficient of x^3 will be the coefficient of x^1 in the first bracket multiplied by the coefficient of x^2 in the second bracket...

A company makes digital clocks. It is determined that 5% of all clocks produced are defective. you go to the warehouse and randomly select 80 clocks. 1. How many of the 80 clocks do you expect to be defective? 2.What is the probability that exactly 6 of the clocks are defective? 3. What is...

In the general expansion of (1+x)^n what does the sum of the coefficients mean?

Suppose that ##a##, ##b##, and ##x## are integers. How would the ##±##s be correctly assigned in such an equation?

Why tossing a coin three times is said to have binomial distribution? I'm little bit confused.

In a game heroes have a maximum dodge rate, from experimental data we have 13 dodges out of 24 attacks (so 11 hits). A fellow on my discord server had immediately solved for the dodge rate as being 13/24. I started to explain it is not so simple as dividing (24-11)/24=13/24 is not the dodge...

a) since np has to be greater than 5, n*p= 50*.5 =25 so yes, we can use this since it is much larger than 5. now, for mean, i believe the equation is saying that the mean is np, which is 25 but in this equation we do not have a q value, so this is where my issue begins... what should i use...

Suppose our process has a 85% chance of 2 trials and a 15% chance of 3 trials, and the rest is straightforward binomial distribution, do I take the weighted average of the binomial distribution at 2 and 3 trials? This is for a game so, yeah thanks.

Summary: Can someone give me a basic high level overview on how to do a binomial expansion? I'm studying for my E&M test and going over multipole expansion. I'm particularly confused about these lines (Griffiths E&M 4th Edition) 𝓇^2_{\pm} = r^2 \left(1\mp \frac{d}{r} \cos\theta +...

I would like to generate (X,Y) pairs such that they would follow a distribution something like this: This is the sum of three normal distributions. Each distribution could have a different taper along the X and the Y, plus an offset along X and/or Y. So the parameters of these three...

As titled, been cracking my head over it. Thanks in advance!

Please see attached image. I'm not sure why I'm getting this error because this is the format I have used to write programs in Maple before. Any ideas? I'm new to this so not sure how to independently trouble shoot or problem solve this, Thanks!

Hi. Is the binomial theorem ##(1+x)^n = 1+nx+(n(n-1)/2)x^2 + ….## valid for x replaced by an infinite series such as ##x+x^2+x^3+...## with every x in the formula replaced by the infinite series ? If so , does the modulus of the infinite series have to be less than one for the series to...

I found the first 4 terms of the series: ½-(1/16)x^2+(1/64)x^4-(7/1536)x^6. I cannot however simplify this to a sum. the 7 in the numerator of the last term of the above expansion is the sticking point.

Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...

Basically, the way I did this problem was to use a table with a known ##n## and ##k##. In this case, I fixed ##n=5##, and ##k=3##. I wanted to find the number of terms with three ##x##'s and two ##y##'s. I labeled each ##x_i##, ##1\leq i \leq 5##; the ##y_i## are labeled the same way. Anyway...

Prove the binomial identity: $$\sum_{j=0}^{n}(-1)^j{n \choose j}=0$$ - in two different ways

Hello there, I'm working on a kinetic theory of mixing between two species - b and w. Now, if I want to calculate the number of different species B bs and W ws can form, I can use a simple combination: (W+B)!/(W!B!) Now, in reality in my system, ws and bs form dimers - ww, bb, wb and bw...

Evaluation of $\displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}$

I am learning binomial theorem now on my long journey to calculus. I noticed that in older textbooks, the binomial coefficient looks like C(n on top,k on bottom) I don’t think that I can display it here and in newer ones,they look like ##\binom{n}{k}## is the old notation outdated?or this is...

Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...

Homework Statement Random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a random variable with uniform (0,1) distribution. What is Var[Y]? Homework Equations E[Y] = np Var(Y) = np(1-p) for variance of a binomial...

Homework Statement A random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a uniform(0,1) random variable. What is E[Y]? Homework Equations E[Y] = np The Attempt at a Solution The key is determining the probability of...

Homework Statement Hi, I have a confusion in knowing Pi in the equations attached. Eq are related to the Topic Discrete Random Variables in the context of Probability lecture I also can't understand what is P(Y=y|Pi)? Homework Equations Eq are attached The Attempt at a Solution I can't...

What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...

Homework Statement Expand ##(1+3x-4x^2)^{0.5}/(1-2x)^2## find its convergence valueHomework EquationsThe Attempt at a Solution on expansion ##(1+3/2x-3.125x^2+4.6875x^3+...)(1+4x+12x^2+32x^3+...)## ##1+5.5x+14.875x^2+42.1875x^3+... ## how do i prove for convergence here?

Homework Statement a) I have to find and expression for sequence of $b_n$ in terms of generating functions of the sequence of $a_n$ $$b_n = (-1)^{n}(n+1)a_0 +(-1)^{n-1}n a_1+...+(-1)2a_{n-1}+a_n$$ with $$a_n = a_{n-1} +8a_{n-2} -12a_{n-3} +25(-3)^{n-2} + 32n^2 -64$$ b) I have to use the...

Determine an expression for f(x) =(1+x)(1+2x)(1+3x)…(1 +nx),find f⸍(0) .

Homework Statement 12 non-distinguishable attacks from President Snow land in Panem’s 12 districts in a particular week. Assume the attacks are located randomly, with each configuration of attacks equally likely. What is the probability that some district had more than 1 attack? Homework...

Homework Statement How many combinations of people are there if you choose 4 or 5 from a group of 10? Homework Equations Relies on binomials The Attempt at a Solution binomial (10,4) = binomial (10,6) = 210 But when choosing 5 the answer is binomial (10,5) / 2 = 126 Why do I need to divide by 2?

Hi everyone, I'm currently working through Mathematical Methods for Physicists 7th ed. by Arfken/Weber/Harris and there's one question that's been giving me some difficulty. I would appreciate any feedback if possible. Thanks! Chris Homework Statement The relativistic sum w of two...

Homework Statement Derive the bernoulli binomial distribution.Homework EquationsThe Attempt at a Solution Each bernoulii trial could have only two possible outcomes . Let’s name one outcome as success and another outcome as failure. Let’s denote the probability of getting success and failure in...