# Tag Info

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### How many rolls are sufficient to ensure, with probability 99%, that the sum is greater than 100?

Let $X_1,...,X_k$ be i.i.d. Uniform($\{1,...,m\}).$ Then the distribution of $S_k=X_1+...+X_k$ is given exactly by a recursion proved in this paper$^\dagger$, which expresses this distribution in ...
• 14.7k
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### Sum of iid random variables

Use "pgf"s (Probability generating functions). Here, any $X_i$ has the following pgf : $$g(s)=\tfrac19(4+4s+s^2)$$ It is essential to note that $$g(s)=\tfrac19(s+2)^2\tag{1}$$ The pgf of a ...
• 83k
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### Independent random variables with $X^2 + Y^2 =1$

Example ... $X$ and $Y$ are independent, and both have the scaled Radermacher distribution $$P(X=1/\sqrt2) = P(X=-1/\sqrt2) = 1/2.$$ Then $X$ and $Y$ are not constant, they are independent, but ...
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### How many rolls are sufficient to ensure, with probability 99%, that the sum is greater than 100?

While not a rigorous answer, I don't know how to get the number analytically, this should suffice. Let $X_i$ be the result of the ith roll and $X_{i}\in\left\{ 1,2,3,4,5,6\right\}$ with equal ...

### How many rolls are sufficient to ensure, with probability 99%, that the sum is greater than 100?

One way to do this is to invoke the central limit theorem which says that as you sum more and more i.i.d random variables, their sum get close to a Gaussian. Let's assume that by the time the sum has ...
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### Independent random variables with $X^2 + Y^2 =1$

If $X$ and $Y$ are independent and $X^2+Y^2=1$, then $X^2$ and $Y^2$ are almost surely constant. Indeed, let $U=X^2$ and $V=Y^2$. Then $U=1-V$ and the random variables $U$ and $V$ are independent ...
• 174k
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### Calculating Density Function and Conditional Expectation of Independent Exponential Random Variables

A note on how to "derive" the convolution formula (it is not the most general version). Let $X,Y$ be independent random variables with continuous densities $f_X, f_Y$. Using conditional ...
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• 451k

### Simple question on the meaning of $y$ in $Y = y$

In probability theory we are working with a probability space, that is a triplet $(\Omega, \mathcal{F}, \mathbb{P})$ consisting of: $\Omega$ a sample space. This is a set of all possible outcomes of ...
• 462
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Accepted

### Can't understand expected gain calculation in first price auction

In the statement OP mentioned, $x$ is seen as fixed and $b$ is seen as a random variable (note the sentence "A does not know the value of $b$"). Hence, $x \geq f(b)$ happens with probability ...
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### Is every collection of discrete random variables a function of independent random variables?

Yes, this is always possible. One method would be to take $N = 2^n-1$, and for each $X^i = (X^i_1,X^i_2,\cdots,X^i_n) \in \Omega$ let $p_i := \mu(\{X^i\})$. Let \begin{align*} Y_1 &\sim \text{...
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### If $X\sim N(\mu, \sigma^2)$ and $\Phi$ is the CDF of a standard Normal random variable, what is the distribution of $\Phi(X)$?

Since $\ \Phi\$ is strictly increasing, it has a strictly increasing inverse $\ \Phi^{-1}:(0,1)\rightarrow(-\infty,\infty)\ .$ Then, for $\ 0<x<1\ ,$ \begin{align} P(\Phi(X)\le x)&=P\big(X\...
• 29.2k
1 vote

### Sum of iid random variables

@ABlack's assumption that there should be probabilities in the final sum is correct. Here we show that @ABlack's approach is correct and leads to the same result as @JeanMarie's elegant answer. In the ...
• 109k
1 vote

### discrete random variable probability question

Big thanks for @lulu for the help! the answer to this question is $\frac{\binom{8}{5} \binom{17}{10}}{\binom{25}{15}} = 0.3332$ in my understanding we choose 5 water bottles from the group of 8 using ...
• 111
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### What probability distribution function is this?

This is just my (not very rigorous) summary of the arguments in the comments (community wiki). Let $R$ be the number of runs ($R=3$ in your example). In the original setting we have $L$ as a fixed ...
1 vote
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### Related to Laplace Transform and Expectation operator

You can obtain equation (2) because equation (1) transforms directly into equation (2). Let me walk you through the transformation: First you need to recognize that the expectation of an exponential ...
1 vote

### Finding the joint distribution of two dependent variables $X_1$ and $X_2=(X_1)^2$

There is no joint probability density function for $X_1$ and $X_2$, as they are not jointly continuous; the support of $(X_1,X_2)$ is on one-dimensional segment of $\mathbb{R}^2$ and hence is not two-...
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### Almost surely convergence of Bernoulli distribution ($\frac{1}{n}$)

It's written that $X_n \to 1$ a.s. It's false. Indeed, $X_n \to 0$ in probability and hence $X_n$ can't converge to $1$ a.s. It looks like you imply that $limsup X_n = 1$ a.s. Let us prove it ...
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• 4,054
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### How should independent random variables be distributed to have their product be distributed as a gaussian?

Here is a partial answer inspired by Robert Israel's idea. Actually, computations will turn out to be carried more easily easier with a positive random variable, that is why we will consider the sign ...
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### Continuity of random variable and its CDF relation in $R^d$

$F$ is continuous at $(x_1, \ldots, x_n)$ if and only if $F(x_1-h, \ldots, x_n-h) \to F(x_1, \ldots, x_n)$. The limit of the difference $\lim_{h \to 0} [F(x_1,\ldots, X_n) - F(x_1-h, \ldots, x_n-h)]$ ...
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Only top scored, non community-wiki answers of a minimum length are eligible