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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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28 views

Find density function for $Y = e^X$

$X$ is a continuous random variable. Let $Y= e^X$. Find the density function of $Y$ in terms of density function of $X$. if $X$ is continuous random variable then $Y$ is too. So we know that $$ \int_{...
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1answer
36 views

If $X$ is a random variable, would f(X) and X have a correlation of 1? [closed]

If $X$ is a random, would $f(X)$ and $X$ have a correlation of 1?
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3answers
50 views

Probability of two individuals getting their own hats, after 6 people put their hats on the table.

Two friends, Joe and Enzo, are members of a group of 6 persons who have placed their hats on a table. What is the probability that both Joe and Enzo get their own hats. (5 marks). I am unsure about ...
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2answers
24 views

Digit sums as a possible simple cipher for orienteering

This scenario is a case where I want to covertly confirm to team members a specific GPS coordinate location (like for an orienteering exercise) I have chosen, so they know when they have correctly ...
2
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0answers
20 views

Hoeffding-Type Bounds for Noncentered Variables

Hoeffding's Tail Bound is well-known for subgaussian variables. It can be written in the following way: Assume $X_i$ for $1\leq i\leq n$ satisfies: $$ \mathbb{P}(|X_i-\mu|>t)\leq 2\exp\left(\frac{-...
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11 views

Question regarding determination of density function, if it's given as a function of $x$ multiplied by some constant C.

So, I need to determine whether a function can be a density function of some variable $X$. Do, I simply assume first, that it is and find $C$ by integrating it and equating the expression to 1 and ...
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1answer
17 views

$X$- amount of experiments conducted until certain number of coloured balls is pulled out. Write an distribution of $X$

Given that in the box there are $4$ white and $4$ black marbles and that $4$ marbles are randomly selected and then put back into the box, what is the distribution $X$ if the experiment is conducted ...
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2answers
151 views

Why are all subset sizes equiprobable if elements are independently included with probability uniform over $[0,1]$?

A probability $p$ is chosen uniformly randomly from $[0,1]$, and then a subset of a set of $n$ elements is formed by including each element independently with probability $p$. In answering Probability ...
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1answer
17 views

Joint Distribution of Scaled multivariate Standard normals

Let $X$ be $N(0,I_n)$, a $n$-dimensional multivariate standard normal. Let $Y = X/||X||$. I know $Y$ is distributed uniformly on the surface of a $n$ dimensional "circle" but how do I prove this? The ...
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2answers
65 views

$\mathbb{P}(S<\infty)=0$ where S is the sum of independent RVs $X_1, X_2, \dots$ distributed exp$(\lambda_n)$ with $0 < \lambda_n \leq 1$

Let $X_1, X_2, \dots$ are independent RVs such that $X_n$ ~ exp$(\lambda_n)$ with $0 < \lambda_n \leq 1$ for all $n \in \mathbb{N}$. Defining $S = \sum_{n=1}^{\infty}X_n$ I am trying to prove $...
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20 views

Bias of the researcher's estimate in stratified random sampling , given the true population mean value [closed]

(a) How to calculate bias of the researcher's estimate in stratified random sampling, given the true population mean value Y¯ is 13.667? (b) Likewise, what is the formula for calculating 'unbiased ...
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1answer
13 views

Probability generating function of a Poisson sum of logarithmic-distributed random variables.

This is exercise 5.2.3 (b) from One Thousand Exercises in Probability by Grimmett and Stirzaker: Let $X_1,X_2,\ldots$ be independent identically distributed random variables with the logarithmic ...
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9 views

probability question: seaching a superfair game, and typical game is exponential decline [closed]

From my probability lectures I rember there was a superfair game, where every typical game is exponential decline, but I dont rember the details anymore. superfair means expected gain is greater $0$. ...
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0answers
19 views

Under which equivalent condition a Markov chain has a stationary distribution? [closed]

Let $(X_n: n \in \mathbb{N}_0)$ be a Markov chain on finite (or infinite state space $S$ with the transition probability matrix $P =[p_{ij}]_{i,j \in S}.$ Under what (equivalent) condition the ...
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2answers
34 views

A contestant participates in a game show where three important prizes are offered.

A contestant participates in a game show where three important prizes are offered. His chances of winning the three prizes are $\frac{1}{6}$, $\frac{1}{3}$ and $\frac{1}{2}$, respectively. How many ...
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0answers
22 views

Normal distribution - Marginalization $P(x)$ given $P(z)$ and $P(x\mid z)$

I'm trying to get $P(x)$ given $P(z)$ and $P(x\mid z)$. I tried doing P(x) = P(x|z)P(z) but I didn't manage to arrive at the correct result. $P(z) = N(z\mid 0, I)$ $P(x\mid z) = N(x\mid Wz+ \mu, \...
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0answers
27 views

Expected number of siblings in a family [duplicate]

The question goes as follows: Given a family where the number of children is a random variable $(\mu=1.8, \sigma=0.36)$, if one selects a child randomly, what’s the expected number of siblings the ...
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18 views

Do you know any literature on this mathematical model of aggregation?

I'm currently studying a fun and very simple mathematical model of particle aggregation presented in this 1988 article of Hideki Takayasu, Ikuko Nishikawa, and Hal Tasaki. Here is a presentation of ...
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18 views

Calculate Probability of 3 out of 25 boxes have 1 out of 5 reds [closed]

We have 25 boxes, 5 red balls and 20 green balls. I will pick up 3 boxes in random order. What is the probability that 1 red in 3 boxes? 2 reds in 3 boxes? 3 reds in 3 boxes?
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1answer
58 views

Probability of an event if r out of n events were true.

I need to calculate what is the probability that the next event will be true if r out of n events were true. It is given that the probability for all events is equal and the probability is evenly ...
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0answers
50 views

Probability of being a better game player than player $x$ [closed]

Suppose we have four games and the probability that a player will win the game are as follows: Game 1: 71% Game 2: 55% Game 3: 42% Game 4: 16% The standard deviations for the above percentages are:...
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0answers
57 views

Probability Derivation using Bernoulli Equation

I need help deriving this equation $p_i( \bar{D}=\bar{d} | D=d ) = {i \choose \bar{d}} {N_t -i \choose d-\bar{d}} \pi_1^d (1-\pi_1)^{N_t-d}$ (given equation for derivation with $ 0 \leq \bar{d} \...
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1answer
41 views

Finding conditional expectation of $Y$ given $X$

We select a number $X$ at random from $\{-3 , -1 , 0 , 2 , 5 \}$. If $X\le 0$ then we choose $Y$ randomly from $[X-2 , X+3]$ otherwise from $[-X , X]$ interval. Calculate value of $\mathbb{E}[Y]$. My ...
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2answers
44 views

Expected value of rolling 4,5,6 vs 6,6,6

I think I'm confused about something very basic with the definition of expected value. Consider the question of "what's faster to get on average rolling dice: consecutive $4,5,6$ or consecutive $6,6,6$...
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0answers
25 views

Probability space $\Omega$ with multiple $\sigma$ - algebras

I have a question relating $\sigma$ - algebras. If there is a probability space $\Omega$ then $\sigma$ algebra is a set consisting of subsets of $\Omega$. There can be a $\sigma $ algebra that doesn't ...
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1answer
17 views

Finding probability that the $n$-th ball drawn was red in terms of the expected number of red balls in $n-1$ draws

Assume that an urn contains $b$ blue ball and $r$ red balls. We pick balls from the urn one at a time, and inspect their color. After we inspect the ball, we put it back, and then we add $k$ balls of ...
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0answers
12 views

Probability of an at least n occurences of subsequences of length k in a RNA of length n

Suppose we have a RNA sequence of length N, then we have a subsequence of length K, $K \le N$ Is there a method to calculate the probability of the subsequence occuring an arbitrary number of times ...
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3answers
54 views

Showing $P(X_1+X_2<1)=\frac12$ where $X_1,X_2$ are i.i.d $U(0,1)$ variables

I am given that $X_1$ and $X_2$ are iid $U(0,1)$ and want to show that $$Pr[X_1+X_2<1]=0.5$$ My approach is to evaluate $$\int_0^1\int_0^{1-x_1}1 \quad dx_2dx_1$$ but there seems to be a ...
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0answers
30 views

How can I compute the conditional expected value $\Bbb E(1/X | T)$?

I need to compute $\Bbb E(1/X \mid T)$, where: the distribution of $X$ is the gamma with parameters $(2,m),\quad m>0$; the distribution of $T$ is the gamma with parameters $(2, nm)$.
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1answer
11 views

Probability that at least one of three exam tickets randomly selected will be hard, if student considers 5 tickets out of 20 difficult.

My interpretation: $$A \text{ - an event, that the ticket is difficult}\\ B \text{ - an event, that the ticket is easy}, n \text{ - the amount of tickets, } m \text{ - the amount of difficult tickets ...
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0answers
52 views

Why do the following facts implies that the cov matrix is PSD. Is anyone willing to help?

Let $V=(v_{ij})$ be a covariance matrix associated to the random vector r. My professor writes that the covariance matrix is PSD. Indeed for all $i,j = 1,...,n$ we have $v_{ij} = E((r_i −E(r_i))(...
2
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2answers
47 views

Two players coin flipping

Two players are flipping a coin. If head, player A wins 1 point. If tail, player B wins 1 point. The person who first wins 2 points wins the game. The loser must pay the winner $\$1$. ...
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2answers
22 views

A die is thrown two times. $A$ is an event, where sum of obtained numbers is less than 6, and $B$ - that both numbers are even. What is $P(A\mid B)$?

So, the probability, that event $B$ occurs is $\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$. Probability that $A$ occurs is $\frac{1}{9}$. Probability that $B$ occurs when $A$ has occurred is $P(B \mid A)...
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0answers
46 views

Expected value for Brownian motion E(B1|B2=1)

I am currently stuck on the following question: $\mathrm{E}\left[B_{1} | B_{2}=1\right]$. Is there any one who can assist me? I can't figure out: $\mathrm{E}\left[B_{1} +B_{2} - B_{2}|B_{2}=1 \right]$...
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2answers
56 views

Birthday problem-Probability exactly $2$ triples and $4$ pairs if $20$ people in room

Say there are 20 people in a room. What is the probability there are exactly 2 triples and 4 pairs. Is my answer shown below correct? Assume 365 days in the year. $P= \dfrac{\binom{365}{2}\binom{363}{...
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1answer
18 views

Log of the Gaussian data likelihood

Given that the model distributon is Gaussian with known variance $\sigma^2 = 1$ $$ p(x_i | \mu, \sigma^2=1) = \mathcal{N}(x_i | \mu, 1) = \frac{1}{\sqrt{2\pi}}\exp{-\frac{(x_i - \mu)^2}{2}} ,$$ the ...
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1answer
123 views

Find $\lim_{n\to \infty}\mathbb P (X + n = Y)$ where $X,Y$ are independent Poisson variables

A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. It is known that $X, Y$ are independent random variables, and it is given that $X \sim \mathrm{Poisson}(n), Y \sim\mathrm{Poisson}(2n)$. ...
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2answers
41 views

If $DX=5$ find E((2+X)^2).

So, if DX=5, then $$E((2+X)^2)=E(4+4X+X^2)=E(4)+4EX+EX^2=4+4EX+EX^2$$ $$DX=E(X-EX)^2=E(X^2-2XEX+(EX)^2)=5$$ $$DX=EX^2-2(EX)^2+(EX)^2=EX^2-(EX)^2=5$$ $$EX^2=5+(EX)^2$$ $$E((2+X)^2)=9+4EX+(EX)^2$$ ...
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28 views

Seemingly easy probability that can't be solved using multiplication. [closed]

There are 15 colored beads; 7 are red, 3 are blue and 5 are green. Three beads are selected at random and replaced. Find the probability that one is red, one blue and one green. The answer is 14/75. ...
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0answers
29 views

Let $F$ be an algebra. Prove that $\forall A\in\sigma(F), \varepsilon>0, \exists B\in F\mid\mathbb P((A\setminus B)\cup(B\setminus A))< \varepsilon$. [duplicate]

I tried approaching the problem by defining $S = \{A\in\sigma(F) : \forall\varepsilon >0 \exists B\in F\mid\mathbb P((A\setminus B)\cup(B\setminus A))<\varepsilon\}$ and then showing that $S$ ...
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0answers
10 views

Distribution smoothing and rising/falling probabilities

Say I have a sequence of real numbers $a_1, a_2, ..., a_n$ where $a_i \geq 1$, and $a_i \leq a_j$ if $i < j$. Define $p(i) = \frac{a_i}{\sum_{j=1}^n a_j}$ and $p_\alpha(i) = \frac{a_i^\alpha}{\...
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0answers
28 views

If $X_1,X_2$ are not independent, are $(X_i)_{i=1}^4$ independent given $Y$?

Given random variables $X_1,...,X_4,Y\in\{0,1\}$ such that $X_1$ and $X_2$ are not independent. Do $X_1,...,X_4$ independent given $Y$? i.e. is $$ P[X=x|Y=y]=\Pi_{i=1}^4P[X_i=x_i|Y=y] \\ \forall x=(...
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0answers
11 views

Relationship between mean error and Expectation of absolute difference [closed]

We know that $E(X-c)^2$ is minimized for c = E(X) and that $\sum(x_i-c)^2$ similarly is minimized for $\bar x$. Further we know that $E|X-c|$ is minimized for c= median of X and that $\sum|x_i-c|$ is ...
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0answers
27 views

Probability that a line intersects area created by circles

Assume we have a line and several circles randomly distributed in such conditions: The line intersected all circles. The circles always connected to each other and create a cluster. As you can see ...
1
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1answer
27 views

Limsup of random variable

Let $X_n$ be a non-negative decreasing sequence of random variables i.e. $X_n < X_{n-1}$ for $n\in\{1,2,...,\}$ such that $\mathbb P[0\le X_0<\infty] = 1$. Is $\sup\mathbb E[X_n]<\infty$ ? ...
1
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1answer
44 views

Full House Probability: Why is my answer incorrect?

For the probability of a full house, I did: 52/52 * 3/51 * 2/50 * 48/49 * 47/48 = 3/20825 I was off by a factor of 10 or in other words: ...
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0answers
22 views

Is such a bayesian model possible?

Let $Y \sim Bernoulli(p)$ and $X|Y,p,\mu_1,\mu_2,\sigma \sim Y*N(\mu_1,\sigma)+(1-Y)*N(\mu_2,\sigma)$. Then, using the Bayes formula: $$P(Y|X,\mu_1,\mu_2,\sigma,p) = \frac{P(X|Y,\mu_1,\mu_2,\sigma,p)P(...
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0answers
26 views

How do they get this formula for covariance? [closed]

I am following this video How can you possibly have this? $\textrm{cov}(\Theta ,\Theta +U) = \textrm{cov}(\Theta ,\Theta) + \textrm{cov}(\Theta,U)$ The video says that it because of the: ...
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1answer
23 views

Expected value of pulling the same colored ball 2 times succesively

If there is $20$ balls from which $12$ $red$ and $8$ $blue$ and you draw with replacement until the last two have the same color, what is the expected value of the number of draws needed to succeed? ...
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0answers
55 views

Find the variance of $\underline{x}^{T}A\underline{x}$

I was trying to find the variance of a quadratic form $\underline{x}^{T} A \underline{x}$ where it is given that $$\mathbb E(X_i)=\nu_i, \quad \mbox{Var}(X_i)=\mu_{2}, \quad \mathbb E(X_i-\nu_i)^3= \...