1
vote
1answer
200 views

A urine test, the VMA test

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about 70 % of cases of neuroblastoma. It has been proposed ...
-2
votes
1answer
34 views

A drug treatment [closed]

A certain drug treatment cures 90 % of cases of hookworm in children. Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from ...
0
votes
0answers
42 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
1
vote
2answers
83 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?
1
vote
6answers
54 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 ...
5
votes
1answer
95 views

Proof of Interesting Binomial Identity

In my work I've come across the interesting binomial identity $$ \sum_{n\geq k} \frac{\binom{n}{k}}{\binom{m-1}{k}} \frac{\binom{m-1}{n} \binom{i-m-1}{j-n-1}}{\binom{i-2}{j-1}} = ...
0
votes
1answer
57 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 ...
0
votes
1answer
24 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 $x$ ...
1
vote
2answers
54 views
1
vote
1answer
53 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 ...
4
votes
1answer
28 views

Properties of cumulative binomial distribution

Let $F\left(k, n, p\right) = \sum_{i=1}^k\binom{n}{i}p^i\left(1-p\right)^{n-i}$ denote the cumulative binomial distribution function. If $F\left(k, n, p\right)-F\left(k, n, p'\right) \geq ...
0
votes
0answers
27 views

how to compute the variance of the random variable Y in binomial distribution?

sorry to bother but I just saw some slides provided by the harvard university. One of those show the binomial distribution with the VAR(Y)=$\frac{\pi(1-\pi)}{N}$ Im bit confused because usually I see ...
1
vote
1answer
118 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 ...
2
votes
1answer
59 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
0
votes
1answer
146 views

Conditional binomials

I'm trying to proof this, If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distribution Y ~ B(n, pq) . Can someone show me how or link some reference.
0
votes
2answers
37 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
32 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 ...
1
vote
1answer
1k 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
48 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
98 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 ...
0
votes
2answers
339 views

Multivariate Hypergeometric Distribution With Wildcard

In the wikipedia link http://en.wikipedia.org/wiki/Hypergeometric_distribution it is obvious you can do things like "I draw 5 cards from a deck of 50 cards where there are 10 cards that equate to ...
3
votes
2answers
737 views

Binomial theorem in probability

We know according to binomial probability theorem , $$P= \binom{n}{r} p^r (1-p)^{n-r} \tag{1}$$ Now If I flip a coin 10 times and want to get the probability for 4 heads then we get according to the ...
0
votes
2answers
49 views

standard deviation of a certain distribution

If I have a list of N outcomes of drawing a number from the set {-1\$,+1\$}, and I know that the probability of getting (in a single draw) (-1\$) is p, and probability of getting (in a single draw) ...
-1
votes
1answer
48 views

Formula issues when working out chances of getting certain marks [duplicate]

$$P(X = k) = \binom{N}{k} (0.5)^k (0.5)^{N-k} = \binom{N}{k} (0.5)^N$$ Using formula above, I have got the following results for chances for getting certain percentage on a $50$ question paper, each ...
0
votes
3answers
159 views

How to correctly write a binomial distribution for a $50$ questions exam [duplicate]

Using binomial distribution I want to know what is the chance of getting $70\%$ or greater in a $50$ question exam, each question having a true/false option to select from. What is the correct formula ...
0
votes
1answer
2k views

Probability generating function of the negative binomial distribution.

I am using the definition of the negative binomial distribution from here. This is the same definition that Matlab uses. For convenience, $$P(k) = {r + k -1 \choose k}p^r(1-p)^k ,$$ where $p$ is ...
1
vote
0answers
46 views

is this binomial distribution correct?

I am trying to work out what is the chances of getting the following marks $(100\%, 70\%, 60\%, 50\%)$ in a paper containing $50$ questions, each question containing yes/no options. Using Binomial ...
2
votes
2answers
68 views

Calculating an “at least” probability without summation?

I know One can calculate the probability of getting at least $k$ successes in $n$ tries by summation: $$\sum_{i=k}^{n} {n \choose i}p^i(1-p)^{n-i}$$ However, is there a known way to calculate such ...
1
vote
2answers
65 views

The probability of at least two persons of $6$ boys and $8$ girls live for $80$ years

I have this small problem, that has been preventing me from proceeding in my studies. I'd be very grateful if someone could help me out. "$29.4$% of boys and $55.1$% of girls are expected to live for ...
2
votes
1answer
299 views

Probability involving cards, using negative binomial and hypergeometric distributions

I'm working on the following problem, and I don't really know how to approach it! It has 3 subsets: You randomly take one card at a time from a deck, with replacement. How many 'draws' are needed to ...
9
votes
4answers
2k views

Expected Value of a Binomial distribution?

If $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom ...
1
vote
0answers
93 views

Recognizing binomial mixtures

I'd like to know a procedure to recognize whether a given probability distribution over outcomes $\{0, \dots, n\}$ can be expressed as a mixture of $n$-trial binomial distributions with different ...
1
vote
0answers
31 views

For $X \sim \mathrm{Binomial}(n,\frac{1}{2})$ does there exist $a,b,c,Y$ s.t. $\Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$?

I need to upper bound some complicated expressions involving binomial distributions: Let $X \sim \mathrm{Binomial}(n,\frac{1}{2})$. I want to find $a,b,c,m$ such that for $Y \sim ...
2
votes
1answer
249 views

Going from binomial distribution to Poisson distribution

Why does the Poisson distribution $$\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k \exp{(-\lambda})}{k!}$$ contain the exponential function $\exp$, while its relation to the binomial distribution would ...
3
votes
2answers
138 views

Combinations and Gaussian function

I notice that the function $\binom{C}{x}$, where $C$ is some constant, resembles a Gaussian function; for example, here is the plot for $\binom{20}{x}$: This corresponds to the Gaussian function $a ...
1
vote
1answer
45 views

Distribution of number of special elements chosen: m choices of n items with k special items

Suppose I a set with $n$ items, $k$ of which have a certain property($k\leq n$), and I choose $m$ items randomly from that set($m\leq n$), what is the distribution of the number of chosen items having ...