Binomial Definition and 640 Threads

Homework Statement Homework Equations The Attempt at a Solution

Homework Statement I'm sorry this doesn't look too nice but it is supposed to be two matricces. Show: |1 a1-b1 a1+b1| |1 a1 b1| |1 a2-b2 a2+b2|=2*|1 a2 b2| |1 a3-b3 a3+b3| |1 a3 b3| without evaluating the determinants. Homework Equations...

Homework Statement Find the term with the specified power in the expansion of the given binomial power. \left( {x^3 + y^2 } \right)^{42} ,\,\,\,\,\,y^{15} Homework Equations {\rm{term}} = \frac{{n!}}{{r!\left( {n - r} \right)!}}x^{n - r} y^r The Attempt at a Solution...

Homework Statement I am doing a poof and I need to use the binomial theorem. However is the following a correct way to rewrite it? (a+b)^n\ =\ {n \choose 0}a^{n} + \sum_{k=1}^{n}{n \choose k}\ a^{n-k}\ b^{k} Homework Equations (a+b)^n\ =\ \sum_{k=0}^{n}{n \choose k}\ a^{n-k}\ b^{k}...

Homework Statement Prove that \sum\limits_{k=o}^l {n \choose k}{m \choose l-k} = {n+m \choose l} Hint: Apply binomial theorem to (1+x)^n * (1+x)^m Homework Equations The Attempt at a Solution Using the hint, I started by saying that (1+x)^n * (1+x)^m = (1+x)^(n+m) =...

Homework Statement Hi all, just a quick question here - the setup is as follows: X is a random variable, X \sim \operatorname{Bin}(m,p) where p=2^{-\sqrt{\log n}}(\log n)^2 and m \geq 2^{\sqrt{\log n}}c for constants c, n (n "large" here). I wish to show that \mathbb{P}(X < c) \leq e^{-(\log...

Homework Statement Use the binomial theorem to rpove that for n a positive integer we have: (1 + 1/n)^n = 1 + sum(k=1 to n) [1/k! product(r=0 to k-1) (1 - r/n)] The Attempt at a Solution (1 + 1/n)^n = 1 + sum(k=1 to n) (n choose r) 1^n-k (1/n)^k, where (n choose r) = n!/r!(n - r)...

Homework Statement Show that the generating function A(x) = \sum_n a_n x^n of a_n = \sum_{k=0}^n {n+k \choose 2k} 2^{n-k} satisfies A(x) = \frac{1-2x}{4x^2-5x+1}Homework Equations The Attempt at a Solution A hint was given to "interchange the sums". After doing that, I don't see how to...

I might need you guys to help me see how this proces, will be distributed: Suppose we have a large amount of elements N(≈1012). I'm simulating a system where I for each iteration damage a random element. If an element gets damaged its damagecounter goes up 1. So say I pick element number...

Suppose you have a coin with 4 fair sides, flip it 5 times, and want to know the probability of 5 heads. This is K(10,5) * (0.25)5 * (1-0.25)5 = K(10,5)*0.255*0.755 Or more generally for any binomially distributed outcome: 1) p(x=r) = pr*(1-p)n-r*K(n,r) But also we must have that: 2) p(x=r)...

Is the binomial distribution, what you call a product distribution? How can I see that, if that is true?

Homework Statement A quality control engineer tests the quality of produced computers. Suppose that 5% of computers have defects, and defects occur independently of each other. A- What is the expected number of defective computers in a shipment of twenty? B- Find the probability of exactly...

Hi there, I'm looking at a problem and wanted some help to advise if I'm going in the right direction. I need to test if the number of times a lotto ball has appeared in a draw fits a binomial distribution. I have collated the data and ultimately will do a hypothesis test. The draw...

Is quite easy to understand. What I don't understand though is this: When you sum over all the binomial probabilities from i=0 to n you should get 1, as this corresponds to the total probability of getting any outcome. I just don't understand what it is, that guarantees that you always get one...

Part I. Write out the binomial expansion for each binomial raised to the 8th power. 1. (x + y) 2. (w + z) 3. (x - y) 4. (2a + 3b) Part II. Now explain how your answer for #1 could be used as a formula to help you answer each of the other items. In each case, for #2, 3 and 4, tell...

Homework Statement Show that if the greatest term in the expansion of (1+x)2n is also the greatest coefficient, then x lies between n/n+1 and n+1/n. Homework Equations No idea. The Attempt at a Solution Don't know where to start.

Homework Statement I'm asked to find (a/b) in the simplest form if the co-efficient of x^8 is zero in the expansion of: (1 + x)(a - bx)^12 Homework Equations Binomial expansion formula ... (a + b)^n = Sum of r --> n (r = 0) (nCr)(a^(n-r) * b^r The Attempt at a Solution...

I have to determine the coefficient of an x term in an expansion such as this; Determine the coefficient of x^18 in the expansion of (1/14 x^2 -7)^16 The general term in the binomial expansion is nCk a^k b^(n−k) I could let a = (1/14 x^2) b = -7 n = 16 k = 9? I have no real idea of how to go...

1. Homework Statement (a) Calculate the electric field at an axial point z of a thin, uniformly charged cylinder of charge density ρ , radius R, and length 2L. z is the distance measured from the center of the cylinder. (b) What becomes of your result in the event z >> L ? 2. Homework...

Hello, so suppose we have B(n,p), where n is discretely uniformly distributed on the integers of the interval (1,5) Is the expected value 3p, and is the variance 3p-p^2 ? I arrived at those answers by treating n as another variable, so np/5 summed over all n is 3p, and similar logic for...

Homework Statement Let n be an element of the positive numbers (Z+). Prove that 3 divides (3n n) or "3n choose n". Use the definition of a binomial coefficient to solve. Homework Equations Definition of a Binomial Coefficient: (n k) := ( n! / k!(n - k)! ) The Attempt at a Solution...

Homework Statement http://puu.sh/epl6 Answer http://puu.sh/eplm Homework Equations The Attempt at a Solution No clue on how to attempt this problem. Any help would be appreciated, thanks!

Homework Statement Midvale School for the Gifted has two types of students: Guessers and Swots. All Midvale tests consist of sets of questions with yes/no answers. Guessers will simply answer yes or no to each question as the mood takes them, so they have probability 0.5 of getting each...

Homework Statement Two teams, A and B, play a series of games. If team A has probability .4 of winning each game, is it to its advantage to play the best three out of five games or the best four out of seven? Assume the outcomes of successive games are independent.Homework Equations...

Homework Statement http://puu.sh/dOcM Answer: http://puu.sh/dOcZ Homework Equations The Attempt at a Solution I got Part A. For part A, this is what I did: I did Egg A: X ~ (6,(1/6)) P(X = 1) and did something similar for Egg B. I then multiplied both to get the answer for Part...

Hey, there's this thing I can't wrap my head around. Let's say we have a negative binomial variable x, with parameters p and r. That is, x is the number of failures we get before the rth sucess, while looking at random bernolli variables with sucsess rate p. It can be shown that...

I'm having a bit of trouble understanding a probability distribution of 2 variables. Take for example taking n cards from a deck, and seeing what is the probability of getting X queens and say Y aces (with replacement). This involves the binomial distribution. The probabilities for the...

Homework Statement Let a be a fixed positive rational number. Choose(and fix) a naural number M > a. a) For any n\inN with n\geqM, show that (a^n)/(n!)\leq((a/M)^(n-M))*(a^M)/(M!) b)Use the previous prblem to show that, given e > 0, there exists an N\inN such that for all n\geqN, (a^n)/(n!)...

NOte this is not a homework nor related to any course nor any test problem etc. - entirely my own interest and study. Re\: text by Biedenharn and Louck "Angular momentum in Q.Physics" . I derive an expression for the norm squared wrt a certain expression in Boson calculus. You don't really...

Homework Statement Use the above to prove that given a rational number a > 1 and A any other rational number, there exists b ε N such that ab > A. Homework Equations The above refers to the proving, by use of both induction and binomial theorem, that (1+a)n ≥ 1+na. Binomial Theorem: (i=0 to...

The coefficient of x in the expansion of [x+(1/ax^2)]^7 is 7/3. Find the possible values of a. 1. Rewrite (x + 1/(ax^2))^7 = x^(-14) (x^3 + 1/a)^7. So, we need to find the coefficient of x^15 from (x^3 + 1/a)^7. 2. Using the Binomial Theorem, we have (x^3 + 1/a)^7 = Σ(k = 0 to 7) C(7...

Homework Statement Prove that \sum_{k=0}^n {3k\choose k}\ge \frac{5^n-1}{4}Homework Equations {3k\choose k}= \frac{(3k)!}{k!(2k)!}The Attempt at a Solution I tried using the induction principle, but... Here my attempt: For n=0 1>0 ok Suppose that is true for n, i.e.: \sum_{k=0}^n...

Product Testing A supposed coffee connoisseur claims she can distinguish between a cup of instant coffee and a cup of drip coffee 75% of the time. You give her 5 cups of coffee and tell her that you will grant her claim if she correctly identifies at least 4 of the 5 cups. (a) What are her...

In a 22-item true–false examination, a student guesses on each question. If 14 correct answers constitute a passing grade, what is the probability the student will pass? i did c(22,14)* (1/2)^14 * (1/2)^8

A fair coin is tossed 5 times. What is the probability of obtaining exactly 2 heads if it is known that at least 1 head appeared?

Hi, An airline knows from past experience that the probability of a person booking a seat and then not turning up is 0.04. A small plane has 50 seats and 55 bookings are made. a) A binomial distribution is used to model this situation. What assumption must be made? Comment on how...

Hi, I am trying to understand the binomial theorem, and would appreciate any insight or pointers. To make notation simpler I'll call the binomial coefficient f(n,k). I understand the combinatorial argument that f(n,k) = f(n-1, k-1) + f(n-1, k). This is, to my understanding, a two...

If X is a binom. rand. var., for what value of θ is the probability b(x;n,θ) at max? Ive no idea... My only guess (most likely wrong) is that max and min are always derivatives... So do i just differentiate and express θ...? Any suggestions...?=( Thank you!

Homework Statement Find the MGF (Moment generating function) of the a. geometric distribution b. negative binomial distribution Homework Equations geometric distribution: f(x)=p^x(1-p)^{x-1} where x=1,2,3... negative binomial distribution: f(x)= \frac{(x-1)!}{(x-r)!(r-1)!}p^r(1-p)^{x-r}...

what will be the coefficient of the x^n in the expansion of (1+x)^-1 and(1-x)^-1. Please answer it separately..

In sending 10 characters, a character error occurs independently with probability 1/10. What is the probability that in a 10-character message, less than 3 errors occur? I am using the binomial formula (n choose k)pk(1-p)n-k where n = 10, p = 1/10, and k is the number of errors. Since the...

Homework Statement Homework Equations The Attempt at a Solution I am really stuck, I have no clue how to even begin. For part B I tried changing the RHS to factorials but I was left at a dead end there.

Homework Statement https://www.physicsforums.com/attachment.php?attachmentid=39642&stc=1&d=1317853920 how do you go about solving this? Homework Equations i have proved the binomial theorem.The Attempt at a Solution i was considering cases, for j(even or odd). would this be the right direction?

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Homework Statement Consider an ideal gas of N identical particles in a volume V, and a subvolume v. The chance a molecule is in inside the subvolume is P = v/V. a) What is the chance the subvolume contains n particles? b) Use the binomial theorem (p + q)^N = \sum_{n = 0}^N p^n q^{N-n}...

Data is collected on the number of fish caught per day on a month long fishing expedition. It is hypothesised that the data are consistent with a negative Binomial random variable ,X , starting at 0, so that X~Neg Bin(k,p) where E[X]=k(1-p)/p and Var =k(1-p)/p^2 . However, before a hypothesis...

I want to show that the binomial distribution: P(m)=\frac{n!}{(n-m)!m!}p^m(1-p)^{n-m} using Stirling's formula: n!=n^n e^{-n} \sqrt{2\pi n} reduces to the normal distribution: P(m)=\frac{1}{\sqrt{2 \pi n}} \frac{1}{\sqrt{p(1-p)}} exp[-\frac{1}{2}\frac{(m-np)^2}{np(1-p)}]...

Hi, the following is a list of binomial cumulative distribution of the probability that out of 25 investors, the number of investors that would have exchange-traded funds in their portfolios. We were asked for the probability that at least 14 investors do not have exchange-traded funds in...

Homework Statement (v-4)/(5v+1) The Attempt at a Solution I'm an engineering student, and I'm taking Differential Equations, but I can't remember how to do simple things like this. A walk through explanation would be very much appreciated, I don't have a lot of time to spare. The...

Homework Statement Prove that for all positive integers n, the equality holds: SUM(nCk)*2^k=(3^n+(-1)^n)/2 Note: The sum goes from k=0 to n. AND k has to be even. Homework Equations Binomial Theorem The Attempt at a Solution I know that if we use the binomial theorem for x=2 and...