Tagged Questions

Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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Transformation technique to find PDF

Consider two random variables with the following joint PDF: $$f_{X,Y}(x,y) = \begin{cases} 2, & x > 0, y > 0, x + y < 1 \\ 0, & \text{otherwise} \end{cases}$$ I need to find ...
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Generating samples from a Beta(2,2) distribution

I'm looking for a convenient way to generate $\text{Beta}(2,2)$ random variables, using independent $\text{Uniform}(0,1)$ random variables and elementary functions. I'd prefer to avoid rejection or ...
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Conditional Probability: Birth rank of children in randomly chosen families

(BH 4.7) A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
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Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
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Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
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Queue depth to keep workers busy

I'm trying to find a probability of keeping w workers busy with a q queue depth feeding those w workers. When the queue has at least one item in it the item can be taken and the item was randomly ...
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Understanding the flat (uniform) Dirichlet distribution density over a simplex

This should be really straightforward from the formula, but somehow I'm having trouble understanding the density of a Dirichlet distribution with $\alpha = [1, 1, ... 1] \in R^k$, which is a uniform ...
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Finding $P(S<0)$ with standard Normal Cumulative Distribution function

I know I'm supposed to use the the Standard Normal Cumulative distribution function. But I can't seem to get everything I need. Let $X$ be a random variable with $P(X=-1)=P(X=0)=0.25$ and $P(X=1)=0.5$...
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Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$

$X_{n}$ independent and $X_n \sim \mathcal{P}(n)$ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ? I ...
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Joint density function of $T_1,T_2$ and expectation of $E[T_1 ^2 +T_2 ^2 ]$

Given that $T_1,T_2$ are random variables representing the useful life (in hours) of two electrical appliance. The joint probability function of two variables distributed uniformly in the domain ...
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Distribution of $\lceil X \rceil - X$ where $X$ has an exponential distribution

Suppose $X$ is a random variable with exponential distribution of parameter $\lambda > 0$. That is, $X$ has density $f(x) = \lambda e^{-\lambda x} \mathcal{1}_{\mathbb{[0,\infty [}}$. The question ...
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How to calculate probability distribution of a function of two independent Poisson random variables?

I can't figure out how to determine the probability distribution function of $$aX + bY,$$ where $X$ and $Y$ are independent Poisson random variable. Basically, I want to check whether $aX+ bY$ ...
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Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
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Proving that a positive-integer valued random variable has the lack of memory property iff it has a geometric distribution.

Suppose that $X$ is a positive-integer valued random variable with the lack of memory property which states: Given that $X>n$, then $\mathbb{P}(X=n+k) = \mathbb{P}(X=k)$. Consider the case ...
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Moment Generating Function for $r$th central moment

When using moment generating functions, to find the $n$th raw moment ("$n$th moment about the origin"), you take the $n$th derivative of the MGF and evaluate at $t=0$. To find the $m$th central ...
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Expected value of an infinite sum of random variables

For k=1,2... let Xk be independent and identically-distributed random variables with E(Xk)= $\mu$ and V(Xk)= $\sigma^2$ and let N be independent of the Xk with mean $\lambda$ and variance $\lambda^2$. ...
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Is there any probability model for multi-stage motion of an object such as this.

I have this following case (please refer to attached pic below) where a particle is resting on the ground and it needs a minimum amount of force (Fmin) to reach from one level to the next level. But ...
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Distribution of sum of weighted geometric random variables

Take $g_i$ to be a geometric random variable with parameter $1/2$, such that $$P(g_i = k) = \frac{1}{2^{k+1}}$$ for any integer $i$. I'm surprised at how much more difficult it is to evaluate this ...
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Calculating integral $\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$

This is a follow up question for this question: How can I calculate or simplify the following integral $$\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$$ If I know f(x) is a probability density ...