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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15 views

Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
-2
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0answers
13 views

In tennis, the probability of a player winning a point on serve given serve statistics.

How can I calculate the probability, $p$, of a player winning a point when serving given: The percentage of first serves that the player gets in. (I'm not sure this is relevant/needed). The ...
4
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1answer
36 views

Roll eleven dice such that the product is prime

So the problem is: What is the probability of rolling eleven dice such that their product is prime. The dice is numbered from 1 to 6 and there is an equal chance of getting each number. So in order ...
3
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1answer
25 views

Probability and cards

A box contains 900 cards enumerated from 100 to 999 (Each number appears once and just in one card). I took some random cards without looking at them and calculated the additions of the digits in each ...
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2answers
13 views

Probability Multivariate Distributions

A computer generates two independent fixed numbers from a uniform distribution on the range [0,100].Calculate the probability that the first fixed number exceeds the second by at least 20. I'm ...
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1answer
48 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
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1answer
12 views

Finding the y-coordinate of the peak in a gaussian distribution?

First off all, my general understanding of gaussians is not very good, and I'm having issues getting my head around this because I cannot find an explanation of them I can understand. I'm working ...
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1answer
10 views

Maximizing Varience of Independent Random Variables [on hold]

Suppose X and Y are independent mean 0 random variables, with positive variances a and b, respectively. Find the value of c that minimizes the variance of cX+(1-c)Y?
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2answers
32 views

Showing that the Lindeberg CLT Condition Holds

Suppose we have a sequence of random variables, $\{X_{n}\}_{n\geq 1}$ satisfying: $\mathbb{P}(X_{j} = 2^{j}) = \mathbb{P}(X_{j} = -2^{j}) = \frac{1}{2}$ Then is it true that the CLT holds? Or ...
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0answers
10 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
0
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2answers
29 views

Help me find $P(A \cap B')$ under the given conditions

I was assigned the task to solve this problem by mathematics teacher which I can't solve because it doesn't make sense to me (I think that it is impossible to solve it). Problem: Events $A$ and ...
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0answers
9 views

Random variables set representation in the sample space

Consider that I have two Random variables $ X : \Omega \rightarrow \mathbb{R} \space , Y : \Omega \rightarrow \mathbb{R}^d$ belonging to the same sample space and a measurable function $\space f : ...
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0answers
21 views

Inequality with poisson distribution [on hold]

Let $r>1$ and $X \sim Poiss(\lambda)$. Prove that $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ Does this inequality hold for $r>0$ ?
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0answers
24 views
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0answers
22 views

Random variable: $X\sim Normal(m, {\sigma}^2)$, find the characteristic function of $X^2$

Is it possible, knowing that $X$ is a random variable with normal distribution( with parameters $(m, {\sigma}^2)$), to find the characteristic function of $X^2$ ? What I thought is: Since: $\phi(X) ...
3
votes
1answer
49 views

Probability - Poisson arrival of rain

I'm trying to solve this Poisson problem. A rain shower lasts 10 minutes and in that time deposits $10^6$ raindrops over 100 $m^2$. a) What is the probability of at least one drop landing in 1 $cm^2$ ...
0
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0answers
15 views

Serial Number in a Geometric Distribution

I won't bother posting the whole exercise.Basically, we've got 2000 pc's and 12 of them are malfunctioning. At some point, the exercise writes: We choose the pc's until we find a malfunctioning ...
1
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1answer
21 views

Determine the probability density function of…

Let $X$ be a random variable with normal distribution with parameters: $$m = 1$$ and $$\sigma = 2$$ How can the probability density function of $$Z = -\frac{\ln |X|}{3}$$ be determined?
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1answer
10 views

Conditional Probability of Poisson Variables

I have two independent Poisson variables $X$ and $Y$ with parameters $\lambda$ and $\mu$, respectively. I defined $Z=X+Y$ and found that $Z$ is also Poisson-distributed with parameter $\lambda + \mu$. ...
1
vote
1answer
26 views

Probability mean,variance and standard deviation formula confusion.

I have a confusion in the formula attached. Why and how are the two formulas equivalent ? sigma in the image is the standard deviation of a distribution...
1
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1answer
22 views

About the equivalence of two asymptotic probabilistic statements

Let $g(n)$ be some monotone increasing function of naturals, and let $X_n$ be a sequence of positive random variables. Consider the following two claims: Claim 1. $\exists f=o(g(n)),\ ...
0
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0answers
15 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
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3answers
65 views

Can the probability of an event be an irrational number?

I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number.
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0answers
17 views

Find distribution of rv X_N where N is independent rv and each X_i~exp(\lambda_i)

First time attempting to use MathJax... Excuse my messy question. Question reads: Let $X_1,X_2,\ldots,X_n$ be independent random variables such that $X_i\sim\exp(\lambda_i)$ such that if $i\neq j$ ...
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1answer
35 views

Proof of infinite monkey theorem.

I was just wondering, does the infinte monkey theorem also has a proof? And why is this called a theorem? It is sheer common sense. And what are its applications. I have heard about PHP and IEP and I ...
0
votes
1answer
17 views

Width of Gaussian distribution from N trials of coin tossing

What is the width of the Gaussian distribution that is generated from performing $N$ trials of coin tossing? Example: In a trial of 1000 tosses of a coin, $P(H)=0.5$ and $P'(H)=0.5$, where $H$ refers ...
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0answers
12 views

Sample complexity of coin bias problem

I am reading a paper involving learning in Multi-armed bandit case (its okay if you don't know what that is. Just trying to give context here.) To give sample complexity lower bound, they reduce their ...
2
votes
2answers
47 views

Find the probability of solutions of an equation.

Let $x+y+z=20$. What is the probability that all the solutions are distinct? (No two variables have the same value). Assuming that the solutions are only positive integers or zero. I have tried- ...
0
votes
1answer
15 views

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$, $U = X+Y$. What are $E[X|U], E[Y|U]$?

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$. X, Y are independent. $U = X+Y$. What are the values of $E[X|U], E[Y|U]$? I understand $E[X|U] + E[Y|U] = U$, but I'm not sure how to move ...
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0answers
48 views

REALLY tricky Probability question [on hold]

Here is a board game. $$ \longleftarrow \text{left} \qquad\qquad\qquad \text{right} \longrightarrow$$ $$\bigg| \text{win} \bigg| -2 \bigg| -1 \bigg| \text{start} \bigg|\ 1\ \bigg|\ 2\ \bigg| ...
0
votes
2answers
21 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
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2answers
33 views

Adding probability of multiple dice rolls

Can anyone tell me what are the odds that stage 4 will be reached?: Stage 1: roll a 20 sided die results must be 13 or lower Stage 2: roll a 20 sided die results must be 13 or lower Stage 3: roll ...
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0answers
8 views

Multi-step probability problem. Noise and Stochastic Processes. [on hold]

Please see the image below! I am having issues with this problem and would love a solution.
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0answers
18 views

Geometric Distribution

The police have stated that 20% of the items sold by pawn shops in the city have been stolen. Ralph has just purchased 4 items from one of the city’s pawn shops. Assuming the official is correct, and ...
-1
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1answer
13 views

Find marginal probability distribution of $ X$?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X= 5\ mL,\ \sigma_Y= 2\ mL, \ \mu_X= 120\ mL, \ \mu_Y= 100\ mL$ and $\rho = 0.6.$ How do I find the marginal probability distribution of ...
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0answers
18 views

Integration - probability

Calculate expected value and variance of the uniform distribution on [8; 42]. I have tried inserting the numbers into the appropriate integrals but I could not come to a solution.
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2answers
18 views

Covariance between $X$ and $Y$ of a bivariate normal distribution?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X$= 5 mL, $\sigma_Y$= 2 mL, $\mu_X$= 120 mL, $\mu_Y$= 100 mL, and $\rho$ = 0.6. How do I find the covariance of $X$ and $Y$? I know the ...
0
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0answers
28 views

Teacher for Semi-Blind kid, Conditional expectation and bayes theorem.

I have an interesting question that I came across. I know that this uses Bayes Theorem, but I am stumped in terms of minimizing the expected squared error. This question is nothing I've ever seen ...
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0answers
22 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
2
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1answer
39 views

Showing $E[X_{n+1}|X_1,…,X_n] = a_0+\Sigma_{k=1}^n a_kX_k$

$X_1,...,X_n,X_{n+1}$ are jointly distributed with a Gaussian distribution. We let $X^* = E[X_{n+1}|X_1,...,X_n]$. Show that there exists constants $a_1,...,a_n,a_{n+1}$ such that $X^* = ...
1
vote
1answer
10 views

Negative binomial distribution pmf derivative

this is what is in my probability book: Let $X$ be the number of independent Bernoulli trials, each with success probability $p$, up to and including the $r$th success. $X$ is a discrete random ...
0
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0answers
9 views

How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
0
votes
1answer
19 views

Bitstring Probability

I am not understanding how to apply n choose r and permutations to the following problem. Given a bit string of length 8 that has exactly three 0's, what is the probability that the bit string will ...
1
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0answers
7 views

How to prove exchangeability for a renewal process of inter-arrival times

By definition we have that $X_1, \ldots , X_n $ are exchangeable if $X_{i_1}, \ldots, X_{i_n}$ has the same joint distribution as $X_1, \ldots , X_n $ whenever $i_1, \ldots,i_n$ is a permutation of ...
1
vote
1answer
35 views

Interchange of the expected value and infinite summation $E(\sum_{m=0}^\infty (it)^m Y_t^m/m!)=\sum_{m=0}^\infty E((it)^m Y_t^m/m!)$

Let $Y_t$ be a random variable (Not positive necesarily). Can I make the next assumption? $$E(\sum_{m=0}^\infty (it)^m Y_t^m/m!)=\sum_{m=0}^\infty E((it)^m Y_t^m/m!)$$ Thanks! I think it is correct ...
0
votes
2answers
27 views

Expected value of key presses

I came across this extremely strange problem that revolves around a piano. I'm not sure how to go about solving it because of the peculiarity. There are 9 notes on a mini piano, numbered from 1 to 9. ...
-1
votes
1answer
26 views

A Problem Distribution Function

If I have a probability density function like this $w(x) = 1 - |x| $if $|x| \leq 1$ or $ w(x)=0$ if $|x|\geq 1$, what's the value of the distribution function F(x)? I mean that I calculated ...
0
votes
1answer
24 views

Significance level for a hypothesis test for a linear regression

Consider linear regression model $Y_i=a+b\cdot x_i+\epsilon_i$, $i=1,2,3,4,5$, where $a,b\in\mathbb{R}$ are unknown and $x_1=x_2=1,x_3=3,x_4=x_5=5$, $\epsilon_i$ are iid, normally distributed with ...
0
votes
1answer
27 views

if $X$ is a constant discrete random variable, then so is $E(X)$

show if $X$ is a constant discrete random variable, then so is $E(X)$. I'm asuming the proof is trivial, but I am having torubles understanding what $X$ being constant means? So suppose $X = c$ then I ...
2
votes
1answer
22 views

What is the pdf and cdf of $aX^2+bX$?

If $X$ is normally distributed, $X \sim N(0, \sigma) $, what distribution is $aX^2+bX$? Is there any way to express the cdf and pdf? Thanks.