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 experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.
Prove this using this identity:
k\binom{n}{k}=n\binom{n-1}{k-1}
\binom{n}{1}-2\binom{n}{2}+3\binom{n}{3}+...+(-1)n-1\binom{n}{n}
I was able to do this via differentiation, but not using this substitution. Any hints would be great.
\sumk=0n\binom{n}{k}2=\binom{2n}{n}
Could someone give me a hint as to how to start this. I'm not sure how to really interpret it.
(n-k)\binom{n}{k}=n\binom{n-1}{k}
Right Side: Suppose you create a committe from \binom{n}{k} , then to pick a leader who isn't in the committee but...
\sum_{m=k}^{n-k}\binom{m}{k}\binom{n-m}{k}=\binom{n+1}{2k+1}
I'm not sure how to prove it, I understand the combinatorial proof..i.e. putting it to an example...but i can't derive one side and get the other.
hello,
i am supposed to show that
Sigma of k = 0 to m, (n, k) (n - k, m - k) = 2^m (n, m)
So I have after expanding:
(n, k) = n!/(n-k)!k! and (n-k, m-k) = (n-k)!/(m-k)!(n-m)!
so together the (n-k)! cancels out and I have
n!/k!(m-k)!(n-m)!
and that is
n!/m!(n-m)! which is
(n, m)...
Using the Binomial Theorem and the definition of the derivate of a function
f(x) as f'(x)= lim as h tends to 0 ((f(x+h)-f(x))/h)
Prove that if f(x)=x^n
then
f'(x)=nx^(n-1)
I'm confused as to how to exactly incorporate the nCr "n choose r" into this interpretation of the...
\binom{r}{k}=\frac{r}{r-k}\binom{n-1}{k}
I'm having problems proving this. However, here is my reasoning:
when factoring out an r you get
\frac{r*(r-1)!}{(r-k)!k!}
\frac{r}{r-k}*\frac{(r-1)!}{(r-k-1)!k!}
Is this proper reasoning?
Hi Guys,
I have been given the probability that a drill strikes oil in a region = 0.2.
I know that if I wanted to find the probabilty of say striking oil 3 times out of 5 wells
It would be 5Choose3 = 5!/((2!)(3!)) * (1/5)3* (4/5)2 = 0.0512
My question is how would I go about...
Hi,
My question is based around the idea about calculating the number of possible outcomes when a given number of variables are chosen randomly (all with equal probability of being picked) a given number of times. Most importantly, I an specifically working so that order is redundant. ie AAB =...
Homework Statement
prove, using mathematical induction, that the next equation holds for all positive t.
\sum_{k=0}^n \dbinom{k+t}{k} = \dbinom{t+n+1}{n}
Homework Equations
\dbinom{n}{k} = {{n!} \over {k!(n-k)!}The Attempt at a Solution
checked that the base is correct (for t=0, and even for...
Homework Statement
(a) Expand (1-x6)4
(b) Find the coefficient of xr, where r is a non-negative integer, in the expansion of (1-x)-4 for |x|<1.
(c) Using (a) and (b), or otherwise, find the coefficient of x^8 in the expansion of ((1-x6)/(1-x))4 for |x|<1.
(Answers:
(a)...
Homework Statement
You've got N marbles, and N bins. one by one, a marble is randomly placed in a bin. What is the probability that there will be no marbles in a given bin.
Homework Equations
P=N!/(n1!n2!)*p^n1*q^n2 : binomial probability
The Attempt at a Solution
since we're...
I'm trying to figure out this problem but i keep getting stuck.
Homework Statement
A woman wants to have a 95% chance for a least a one boy and at least one girl. What is the minimum number of children that she should plan to have? Assume that the event that a child is a girl and a boy is...
Homework Statement
The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 70% males and 30% females. The agency calls 30 people chosen at random from its list.
What is the probability that the first woman is reached on the fourth call? (That is, the first 4...
Homework Statement
Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently
a) What is the probability...
Homework Statement
Using the identity (1+2x) (1-2x+4x^2) = 1+8x^3 to expand (1-2x+4x^2)^(-0.5) in ascending powers of x as far as the term in x^3. (The answer is 1+x-x^2/2-7x^3/2+...)
Homework Equations
(1+2x)^(0.5) = 1+x-x^2/2+x^3/2+...
(1+8x^3)^(-0.5) = 1-4x^3+...
The Attempt at...
Hi,
Let's say you start off with $500, and someone offers to give you another $500 everytime a coin is heads, or take $500 from you it's tails. You agree to play this game until a) you've either lost all your money, or b) you've made an extra $1000 (i.e. so you walk away with $1500).
Then...
Hi. i keep finding a different answer than what textbook offers. is my answer correct?
question: the quality control department of a company making computer chips knows that 2% of the chips arw defective. use the nurmal approximation to the binomial probability distribution, with a continuity...
Hello,
Can someone please tell me whether I can use negative binomial distribution for this question.
"If there are 3 types of books in a bookstore and each book has an equal probability of being bought. What is the expected number of purchases to get all 3 books?"
Using negative...
Homework Statement
A coin can be flipped a maximum of four times
The following conditions exist:
H(first) = $1
H(second) = $2
H(third) = $3
H(fourth) = $4
Where H = Heads
And first, second, third and fourth, refer to what order one head is obtained.
What is the expected gain...
Homework Statement
x^2 = 2/11x + 99/121
Homework Equations
The Attempt at a Solution
x^2 = 2/11 x + 99/121
x^2 - 2/11x - 99/121 = 0
x^2 - 2/11x =99/121
I understand that (b/2)^2 must be added to each side to become a perfect square trinomial...But HOW I do it...
I am asked to prove that \sum ^n _{k=0} (C^n_k)^2 = C^{2 n}_n
Where C^n_k signifies "n choose k"
I am told the hint to use the binomial theorem and to calculate the coefficient of x^n in the product (1+x)^n (1+x)^n = (1+x)^{2n}
the Binomial theorem is given by (x+y)^n = \sum_{k=0}...
Homework Statement
Estimate the probability that, in a group of five people, at least two of them have the same zodiacal sign. (There are 12 zodiacal signs; assume that each sign is equally likely for any person.)
Homework Equations
P(X=k) = nCk * p^{k} * (1-p)^k{}
The Attempt at a...
Hi, I've been struggling with this problem for sometime. Let nCk be the kth coefficient in the binomial expansion of (a+b)^n. Find an expression for (n+1)Ck in term of the various nCj. Feel free to treat k=0 and k=n+1 as special cases.
Rudin's proof of lim n-> inf (p^(1/n)) = 1
1+n*x_n <= (1 + x_n)^n = o
I don't see it from the binomial theorem, which is what he says that is from.
He also does things with the binomial theorem like:
(1+x_n)^n >= ((n(n-1)) / 2) *x_n^2
I'm not sure what he did to get these two...
I've started listening to the lectures for the MIT OpenCourseWare 18.01 Single Variable Calculus class. I understood all of it up until the teacher found the derivative of xn. Here's what he wrote on the board:
\frac{d}{dx} x^{n} = \frac{\Delta f}{\Delta x} = \frac{(x+\Delta x)^{n} -...
Hey people, I've racked my brain on this question for hours and can't seem to get to grips with it, wondering if i could get a little guidance?
Homework Statement
Considering the co-efficient of x^n in the expansion of (1+x)^n(1+x)^n, show:
[the sum from k=0 -> k=n of:] [nCk]2 = 2nCn...
Homework Statement
Describe a binomial experiment that can be solved using the expression
10C5 (0.2)5 (0.8)5
Homework Equations
The Attempt at a Solution
Have no idea.
Homework Statement
Find the binary decimal expansion of the fraction 1/3. Identify the repeating decimals of digits.
The Attempt at a Solution
I have that 1/3=0.0101111... and so the repeating digit is 1.
Is this right? It's the first time I've been exposed to binary expansion.
Homework Statement
1. Let Y_1, Y_2, ... Y_n be a random sample from a normal distribution with mean = 2 and variance = 4. How large must n be in order that
P(1.9 <= Y <= 2.1) >= 0.99
(there is a bar _ above the Y in the problem.)
2. When a machine is improperly adjusted, it has a...
Homework Statement
I am trying to figure out if I have a 20% chance to get what I want (let it = x) and I have 6 chances to do so (n=6), I am curious how I set this question up to find out my chances of getting 'x' once out of the 6 times I try.
Homework Equations
Binomial Distribution...
I know how to expand binomials with the aid of pascals triangle and also with the aid of the nCr function on the calculator. I'm not quite sure about this formula though
see the part in the brackets where n is above k. What does that mean? Someone told me that represents nCk. Is that true...
Homework Statement
a) Expand ([2/3]+[1/3])^4
b) Four chocolates are randomly taken (with replacement) from a box containing strawberry creams and almond centres in the ratio 2 : 1. What is the probability of getting:
i) all strawberry creams
ii) two of each type
iii) at least 2...
Ok, so after a little discussion with my discrete math teacher today, he sent me on a little "quest". Here is how it happened:
The topic we were covering was set theory, and as I had been studying very basic combinatorics the night before, I noticed something about the powerset, namely...
Use the binomial expansion of (1+x)^(-1/2) to find an approximation for 1/(rt4.2).
I've got the expansion of (1+x)^(-1/2) as 1-(1/2)x+(3/8)x^2...
but the obvious idea of substituting x=3.2 gives me the wrong answer. I think it's something to do with the expansion being valid but can't...
Ok, the question says
Binomial with n=40. p=0.6 use normal approximation and determine..
a) value at 14.
b)value lest then 12.
So I thought I had this down and packed but the answer in the back of the book tells me otherwise. Anyways the following is my working.
Y~Nor(13.5<x<14.5)...
Homework Statement
A population has an average of 12 defects per 100 feet of wire sampled and inspected. What is the probability of finding 20 or fewer defects in a sample?
Homework Equations
I think I am supposed to use the binomial distribution b(x;n,p)
The Attempt at a...
Hi everyone, I have been having a problem with the General Binomial Coefficient for any rational value:
\left(
\begin{array}{c}
n\\
r\end{array}
\right)
= \frac{1}{r!}\prod_{i=0}^{r-1} (r-i)
Now this works fine except when r=0. so 0! is defined to be 1 so the coefficient of the...
Binomial, Poisson and Normal Probability distribution help!
Hey everyone, I've just started a new section on probability (argh) in my math course, and unlike most other maths, I cannot cope! Anyway, I was wondering if anyone could tell me if I did the following question correctly! Any helps...
Homework Statement
If \sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a, then find the value of \sum^{n}_{r=0} \frac{r}{^{n}C_{r}} in terms of a and n.[/tex]
The Attempt at a Solution
I tried to write down the terms of both the series, but to no avail. i can't think of...
Homework Statement
\sum^{n}_{r=0} (2r+1) (^{n} C_{r})^{2}
The Attempt at a Solution
x(1+x^{2})^{n}
If I differentiate this and put x=1;
I will get the above series without the squares of the binomial coefficients.Will multiplying by (1+x)^{n} help now?
Homework Statement
Evaluate
\sum^{m}_{r=0} ^{ n + r }C_{n}
I can handle things when the lower thing in the combination part is changing, what shall I do with this one?
Homework Statement
If p is nearly equal to q and n>1, show that \frac{(n+1)p+(n-1)q}{(n-1)p+(n+1)q}=(\frac{p}{q})^{\frac{1}{n}}
Note: the index 1/n is on the whole fraction (p/q)
I think it might be helpful if I specify th chapter from which I got this question. Its the binomial thorem...
Question:
Find the coefficient of x^5 in (1+x+x^2)^4.
Problem:
I have not come across expanding brackets which have x^2. I know how to apply the binomial theorem for (a+b)^n or (1+a)^n but have not come across (1 + ax + ax^2)^n. They are not explained in my textbooks so I was wondering if...
http://img529.imageshack.us/img529/3044/bin2od4.th.jpg
http://img529.imageshack.us/img529/9734/bin1ml5.th.jpg
The problem and solution are attached above. Firstly, why is the expansion valid for (mod x/(1+x)) < 1, and not just for mod(x)<1?
Also, I do not understand how from (mod x/(1+x))...