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Questions tagged [probability]

For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].

The probability that an event occurs is a number in the interval $[0, 1]$, which represents how likely the event is to happen. $0$ indicates it will never happen, $1$ indicates it will always happen.

For example, throwing two dice gives a total of $6$ five times out of thirty-six. We write $$P(X=6)=\frac{5}{36}$$.

Use this tag for basic questions about probability, independence, total probability and conditional probability.

For questions about the theory of probability, use instead. For questions about specific probability distributions, use .

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Moments and non-negative random variables?

I want to prove that for non-negative random variables with distribution F: $$E(X^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$ Is the following proof correct? $$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x))…
Osama Gamal
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Derivation of the third moment of Poisson distribution using Stein-Chen identity

(a) Use LOTUS to show that for $X \sim \operatorname{Pois}(\lambda)$ and any function g, $E(Xg(X)) = λE(g(X + 1))$. This is called the Stein-Chen identity for the Poisson. (b) Find the third moment $E(X^3)$ for $X \sim…
NoBackingDown
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Lindeberg's condition vs Lyapunov's condition

Can someone construct an example where Lindeberg's condition holds but Lyapunov's condition does not? This is a problem from Billingsley's/Chung's book. Thank you very much.
Zhanxiong
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Probability brainteaser: expected value of these two games

I found this brainteaser on the internet and do not know how to solve it. For the second question, My first thought is to deduct from the situation when there is only 2 slots, then 3, 4, .., n,.. slots. But I find it not that easy. Can someone give…
Brain Zhang
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Mean and Variance of the Weibull Distribution

The density of the Weibull Distribution is given by: $$f(x) = \alpha x^{\alpha-1}e^{-x^{\alpha}}$$ The Gamma function is defined as: $$\Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x} \,dx$$ Show that $E(X)=\Gamma(\frac{1}{\alpha}+1)$ and…
user137481
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probability of collision with randomly generated ID

Reworded: If 10,000 events occur each second, and each event needs a unique ID, how many random bits does each ID require to insure collisions are minimized? $10,000$ events could occur near the same time, so the time stamp is not sufficient for a…
user168388
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Probability "Brain Teaser"

I came across a "brain teaser" which goes like this: Bruna was first to arrive at a 100 seat theatre. She forgot her seat number and picks a random seat for herself. After this, every single person who get to the theatre sits on his seat if its…
Ben
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Weaker condition for law of large numbers

$X_k$'s are i.i.d. Suppose $X_k$ is symmetric and $E[|X_k|^{3/4}]<\infty$. Do we have $S_n/n \rightarrow 0$ either in probability or almost surely, where $S_n$ is the partial sum.
epsilon
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Understanding conditional expectation and indicator function

$X$ is a random variable which has exponential distribution. We define the event $A=\{a
user10248
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Flip a coin 6 times. What is probability of at least 4 heads?

I can figure out the much simpler case of the probability of getting at least 2 heads in 3 coin flips: There are 8 (2^3) ways to flip a coin 3 times: HHH, HHT, TTT, TTH, HTH, HTT, THT, THH. 4 of these contain 2 or more heads. Therefor the…
NotSuper
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Is there a simple way to illustrate that Fermat's Last Theorem is plausible?

A first step in proving a theorem is true could be to show that it is plausible, so at least you then would have a general idea that it could be true and have something to start with in proving it. Simply put: if you get the picture, you can do the…
Bert Brouwer
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I pull $17$ balls out of a bag, and there are $13$ distinct colors in the sample. About how many colors are in the bag?

I have a bag filled with different colors of balls. My goal is to determine the number of distinct colors that in the bag, but I am limited to taking a small sample. From a sample of $N$ balls, I see that there $X$ different colors. What is the…
PhiNotPi
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Expectation of number of trials before success in an urn problem without replacement

Possible Duplicate: Expected number of draws until the first good element is chosen An urn contains $b$ blue balls and $r$ red balls. Balls are removed at random without replacement until the first blue ball is drawn. What is the expectation of…
wircho
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Expected number of draws until the first good element is chosen

A population has $G$ good and $B$ bad elements, $G+B=N$. Elements are drawn one by one at random without replacement. Suppose the first good element appears on draw number $X$. Find a simple formula, not involving any summation from $1$ to $N$, for…
caligirl11
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Conditional Density Function Derivation

Let $(\Omega, \mathcal{F},P)$ be a probability space and $X\colon\Omega \to \mathbb{R},Y \colon \Omega \to \mathbb{R}$ be continuous random variables (i.e. random variables which have a density function. I am assuming that this implies…
jpv
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