# Tagged Questions

This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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### Conditional expectation of second moment given sum of iid variables.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. Now we want ...
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### CDF of maximum of iid rvs [duplicate]

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know ...
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### A fair dice is thrown six times and the list of numbers showing up is noted. The probability that among the numbers 1 to 6 only 4 nu…

Question : A fair dice is thrown six times and the list of numbers showing up is noted. Now how to find the probability that among the numbers 1 to 6 only 4 numbers appear in the list Please ...
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### create a Gaussian distribution with a customize covariance in Matlab

the Matlab function 'randn' randomize a Gaussian distribution with $\mu= \begin {pmatrix} 0\\0\end{pmatrix}$ and $cov= \begin {pmatrix} 1&0\\0&1\end{pmatrix}$ Ineed to randomize a Gaussian ...
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### There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist-

There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist Entirely of Males? Entirely of Females? 2 males and 3 females?
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### Probability of an integer being a prime

$\Omega=\mathbb{N}^*,P(\omega=n)=\dfrac{1}{2^n}$, let $A_k$ be the event $k\mid\omega$. 1) Find $P(A_k)$ 2) Let B be the event "$\omega$ is prime", show that ...
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### Given a probability distribution, how many times do I have to repeat an experiment so see a certain outcome

My question concerns random number generation under certain constraints. I assume that the random number generator is good enough to generate uniformly distributed numbers. This means that each number ...
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### A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
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### How to analyse a random walk with random transition probabilities

Consider a $1$-dimensional random walk with discrete time steps. We start at the origin and at each integer position there is possibly different probability of moving right one step, or left one step. ...
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### Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
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### Exercise from Norris' book on Markov chains

Let $(X_n)$ be a Markov chain on $\mathbb{N}$ with transition probabilities satisfying: $$p_{0,1}=1,\quad p_{i,i-1}+p_{i,i+1}=1,\quad p_{i,i+1}=\left(\frac{i+1}{i}\right)^{\alpha}p_{i,i-1}$$ The ...
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### Sum of Two Poisson distributions

The probability distribution for the number of goals scored per match by Team A is believed to follow $X \sim Poi(0.8)$. Independently, the number of goals scored by Team B is believed to ...
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### About the definition of mean square convergence.

A sequence of random variables $X_n$ is said to converge to $X$ in mean square if $$\mathbb{E}\left((X_n-X)^2\right) \rightarrow 0 \ \ \mathrm{as\ } n\rightarrow \infty$$. I understand what expected ...
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### Simple Markov Chain: Random Walk on $\mathbb{Z}$

We are given a random walk on $\mathbb{Z}$, where $p_{i, i+1}= p < \frac{1}{2}$ and $p_{i,i-1}=1-p > \frac{1}{2}$, starting at $0$. Now we have to compute the probability that we eventually ...
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### A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$.

A random sample of size 5 is drawn from the pdf $f_{Y}(y) = 2y, 0\leq y \leq 1$. Calculate $P(Y_{(1)} < 0.6 < Y_{(5)})$. (Hint: Consider the complement.) Attempt: The pdf of the largest order ...
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### Convergence in distribution - Proof

I was given a problem: For each $n\in\mathbb N$, let $X_n$ be a random variable with uniform distribution over the set $\{0,\frac{1}{n},\frac{2}{n},\dotsc,\frac{n-1}{n},1\}$. Let ...
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### Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
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### $\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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### Probability of the highest order statistic below the population median. [closed]

What is the probability that the highest order statistic of a random sample of size n from any continuous distribution is below the median ( population median ) of that distribution.
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### Probability: How much days we need to play a game win

Suppose the probability of win a lotery game is : $1/1000$ If a person play the lotery every day with the same combination, how much time he need to wait to win the lotery? Im thinking to use a ...
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### How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
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### Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
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### Interval of probabilities which satisfy a Markov chain

Given the following markov chain, where T1 is the start state, the labels are shown on the state( 'a' in this case) and p and 1-p are probabilities for that transition happening: Now, for what ...
461 views

### Probability of no ace in a 6 card hand, given 4 are not aces.

A player is dealt six cards out of a normal deck of cards. He looks at the first four and notices there is no ace among them. What is the probability that he does not have an ace at all. This sounds ...
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### Find the chance that $a^3 + b^3 \equiv 0 (\mod 3)$

We are given set of integer numbers $\{1,2, \dots N\}$. $N \ge 3$ Then perform a drawing with replacement of two elements $a$ and $b$. Problem is to find the probability of following statement holding ...
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### Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$. I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
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### Basic probability and counting methods

A somewhat geeky problem has been on my mind the last few days: In my accommodation at Uppsala, there are 12 rooms to a floor. I discovered the other day that another British girl whom I know lives ...
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