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Question 2: What would be the best way to interpret this observation? For example does such a shape indicates that a few major/dominant language will eventually cannibalize the numerous minor languages? How many subdistricts are there? It looks like in most districts (about 2 or 3 thousand?) one language is dominant with 80% or more of the people that have ...

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You are correct. $p_X(x)=f(x)$ is just another function and $p_X(Y)$ acts like a transformation on the random variable $Y$, thus making it a random variable.

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You can use Poisson approximation to the binomial, where $p=1/n$, and $\lambda=np=1$. Probability of appearing exactly $k$ times is $$p_X(k)={n\choose k}\left(\frac{1}{n}\right)^k\left(1-\frac{1}{n}\right)^{n-k}\approx e^{-\lambda} \frac{\lambda^k}{k!}=\frac{e^{-1}}{k!}$$

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Yes, basically you've just changed the objects: modem $\rightarrow$ doctor customer $\rightarrow$ patient And, since $n,c,p$ are the same, your probability is $\approx 0.04$ as well.

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The second one is the correct one. The first one is the probability of having the first two draws blue and last draw black. That is different than having two blues and one black in total, i.e. \begin{align}P(\text{2 blue and 1 black}) &= P(Blue, Blue, Black) + P(Blue, Black, Blue) \\&+ P(Black, Blue, Blue)\end{align}

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Some hints: (b) Find $F_X(x)=P(X\leq x)=P(b/U^{1/a}\leq x)=P(U\geq b^a/x^a)$. Take it from here, find $F_X(x)$, and then differentiate wrt $x$ to find the PDF. (c) Pick some $a,b$ of your choice, and using any programming language you like, create several uniform random variables, for each uniform random, calculate $X=b/U^{1/a}$ and get random $X$'s. Plot ...

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The cdf of $T=XY$ conditional on $X=x$ is $F_{T|X}(t)=P(y<t/x)$, which is $1$ if $t>x$ and $t/x$ otherwise. So the density of $T|X=x$ is $f_{T|X}= 1_{[0,x]}(t)1/x$. The conditional density the other way around is $$f_{X|T=t}(x)= \frac{1_{[0,x]}(t) 1/x}{\int_t^1 1/x\,dx}= \frac{-1}{\log t} \frac{1}{x}$$ Given that it's easy to come up with ...

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$$P(2k, n-2k| \Theta) = \frac{2^{n-2k} \binom{\Theta-k}{n-2k} \binom{\Theta}{k}}{\binom{2\Theta}{n}}.$$ Denominator - Unconstrained no. ways to choose socks This is given exactly by the binomial coefficient to choose $n$ objects from $2\Theta$: $$\binom{2\Theta}{n}.$$ Numerator - No. ways to choose $k$ pairs This is again a binomial coefficient: we ...

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I will answer your questions from my experience in learning Probabilistic Graphical Models (PGM) in university and the way my PGM teacher defined probabilistic inference. Knowing the material of this class was based on , I assume you could find more precise answers in this book. In answer to 2: Probabilistic inference is a type of statistical inference. ...

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The region you'll integrate is between the lines $y=x/2, y=x$ and the rectangle $[0,2]\times[0,1]$. You need to draw this to see explicitly. The line $y=x/2$ passes through the upper-right corner of the rectangle and $y=x$ passes to the left of it. You can integrate this region first from $y$ then $x$ or vice versa, but the former is simpler in this figure. ...

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As @gunes (+1) has said: Taken exactly as stated, you have the same question either way. $X \sim \mathsf{Binom}(n=100, p=0.1)$ is the probability a modem/doctor will be busy at a given time. And you seek $P(X > 15) = 1 - P(X \le 15) = 0.0399.$ In R, where pbinom is a binomial CDF, the computation is shown below: 1 - pbinom(15,100,.1)  0.03989053 ...

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