The probability tag has no wiki summary.
0
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1answer
8 views
MGF of random variable
Suppose $X$ has the standard normal distribution and $Y$ has an exponential distribution. How would you find the mgf of $\frac{X}{\sqrt{Y}}$? Would it be $$ \frac{M_{X}(t)}{\sqrt{M_{Y}(t)}}$$
0
votes
1answer
14 views
Probability of random shuffling of cards
I have a pack of cards and use the following method to shuffle them
Pick a random card from the deck and replace the first card with it
Put the first card back in the deck
Move to the second card ...
1
vote
2answers
29 views
Gaussians going towards delta “functions”
We have a sequence of random variables $x_1, x_2, x_3,...$ that are independent and are $N(0, 1/n)$ random variables. We want to show that $(x_1)^2 + (x_2)^2 + (x_3)^2 +...$ converges in probability ...
0
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2answers
15 views
Lower bound on a minimum of maximum of a sequence of standard normal random variables
Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ entries.
We know that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at least $1-\delta$.
I would ...
0
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2answers
25 views
Weighted average of two frequencies
This should be really simple but I'm getting stuck and I'm probably extremely dumb..
I know that a machine receives two kind of parts:
Type 1 -> with frequency 50 per week, each one is processed for ...
0
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1answer
18 views
Probability that a Multivariate Normal RV lies within a Spherical Region of Radius R
I am currently using different procedures to estimate the probability that a $D$-dimensional Gaussian random variable with mean $\mu$ and covariance $\Sigma$ lies within a sphere of radius $R$ that ...
1
vote
1answer
34 views
Card probability problem
I found the following problem in Rosen's Discrete Mathematics and Its Applications 6th ed.:
There are three cards in a box. Both sides of one card are black, both sides of one card are red, and the ...
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0answers
23 views
Distribution of continuous and discrete random variables
I have a question that I encountered while reading my notes.
Suppose we have two independent variables
\begin{align*}
&P_X(dx) = e^{-x} \mathbb{1}_{\mathbb{R}_+}dx \\ &P_Y = ...
0
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0answers
12 views
What methods can I use to calc members needed for doubt/centainty-level
Given a set of percents symbolizing the members individual certainty or doubt.
How can I calculate how many of the members are needed in order for a doubt/certainty-level to be met?
Examples:
Level ...
2
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0answers
24 views
$\sigma$-algebra generated by Brownian motion
Let $(B)_{t \geq 0}$ be a standard Brownian motion. Then $B$ is adapted to its natural filtration $(\mathcal{F}^B_t)_{t\geq 0}$. Often, we want to consider a slightly bigger filtration, ones ...
2
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0answers
40 views
Simple Derivation of Marginalization
So P where P is Probability so P(X) = Prob(X) = P(X|I) = The Probability of Event X occurring:
how can you derive this from P(X) = P(X,Y) $\cap$ P(X,$\bar{Y}$)?
Thank you,
Note: sorry it should ...
1
vote
1answer
27 views
random walk and boundary conditions
Find boundary conditions that allow the recursion $f(k)=\frac{1}{2} f(k-1) + \frac{1}{2} f(k+1)$ for $-B<k<A$ to uniquely determine the function $f(k)=P(S_{\tau}=A|S_o=k)$ where $\tau = \min ...
1
vote
2answers
24 views
Another event probability question
For two events A and B,
P(A) = 0.2,
P(B) = 0.3,
P(~A | ~B) = 0.8
I need to calculate P(~B | A), but I'm not sure if what I'm doing is correct.
So knowing P(~A | ~B), I can say
P(A | ~B) = ...
1
vote
1answer
41 views
Probability of hitting a deep downswing in a sequence
This is really a poker tournament/payout/bankroll question but I think I can frame it in a dice game for easier understanding.
If a player starts with X units, and plays a game where on each throw the ...
0
votes
1answer
34 views
Percentages word problem
There has been an outbreak of bedbugs at a college. 30% of students have the pests in their beds. 40% of the infested beds are in the dorms and 60% are in apartments. Of the uninfested beds, 20% are ...
28
votes
4answers
487 views
Strange Patience Game
I read about this game as a kid, but my maths was never up to solving it:
The score starts at zero. Take a shuffled pack of cards and keep dealing face up until you reach the first Ace, at which the ...
1
vote
1answer
27 views
Tail bound for sum of product Normal-Bernoulli random variables
Let $Z \sim \pi N(\mu, \sigma^2) + (1-\pi)\delta_0$ and $z_i \sim Z$ are iid for $i=1,\ldots,n$. I would like to obtain a result of the form
$$
P[n^{-1}\sum_i z_i - \pi\mu > \epsilon]\leq\exp(-c n ...
1
vote
2answers
27 views
Expectation of Random Variable with even Probability Density Function
By Definition of Expectation of Random Variable:
$$ E(X)= \int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$
Now if the pdf $f_X(x)$ is Even we know that $E(X)=0$ (Ofcourse if integral Converges, i.e, Lets ...
1
vote
1answer
24 views
Handling dropouts in a round-robin tournament.
In a round-robin tournament (where a player versus every other in the tournament in turn), do players who drop out of the tournament after a few rounds affect the probability of winning for those who ...
2
votes
1answer
15 views
The distribution of the sum of two independent log-uniform distribution
I am trying to calculate the distribution of the sum of two independent log-uniform distributions but something doesn't add up.
Suppose $a \sim uni(0,1)$ and $b \sim uni(0,1)$. Thus, $u=log(a)$ has ...
2
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1answer
60 views
Probability Questions
I need help with part (1) of this problem and confirmation I've done the right thing for (2).
1) A survey carried out by a firm found 60% of clients buy products every month and 20% buy high-tech ...
1
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1answer
37 views
dice probability issue
In the game Settlers of Catan, territories/tiles are each (effectively) randomly assigned a number from 2-12. When that number is rolled as the sum of two dice, the tile generates resources for the ...
1
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1answer
35 views
Is this function $f(x) = 2 - \Phi(a-x) - \Phi(a)$ log-concave?
I would like to verify this function $f:\mathbb{R} \to \mathbb{R}$ to be log-concave or determine the region where it is:
$$
f(x) = 2 - \Phi(a-x) - \Phi(a)
$$
where $a > 0$ is a constant, and ...
1
vote
1answer
64 views
Joint Probability $P(X,Y,Z) = P(Y,X,Z)$
Does the order of variables in the joint probability $P(X,Y,\dots)$ have any implications on the statement of joint probability? Concrete, is:
$$P(X,Y,Z) = P(Y,X,Z)$$
To my mind, clearly this is ...
0
votes
2answers
71 views
Sum of 3 loaded dice
I am given 3 loaded dice $D_1$, $D_2$ and $D_3$ and their probability tables $P(D_i = k), 1 \leq k \leq 6$.
I ought to write an algorithm that computes $P(\text{Sum} \mid D_1)$, the sum of all three ...
1
vote
1answer
64 views
Battleship probability matrix
Consider a 10 x 10 Battleship grid that hides a single ship of length = 3.
This ship can be placed vertically or horizontally in any of the 100 cells.
The problem is to get the 10 x 10 probability ...
0
votes
1answer
31 views
MA process ACF proof - don't understand it
I've got the proof but I don't understand a small detail.
As you know for an MA process:
$X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$
where $Z_n$ is WGN (pure Gaussian random process).
Then the ACF is:
...
1
vote
2answers
42 views
Event probability question
The two events E and F have probabilities of $P(E) = P(F) = 0.5$ and they are
dependent since $P(E| \lnot F) = 0.6$. What's $P(E|F)$?
I know Bayes theorem, it's just that I don't exactly know ...
3
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2answers
68 views
probability Theory
I need an explanation on the following problem:
If you are dealt $3$ cards from a shuffled deck of $52$ cards, find the probability that all $3$ cards are picture cards.
I found the total number ...
0
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2answers
43 views
Two-sided Chebyshev inequality for event not symmetric around the mean?
Let $X$ be a random variable with finite expected value $μ$ and non-zero variance $σ^2$. Then for any real number $k > 0$, two-sided Chebyshev inequality states
$$
\Pr(|X-\mu|\geq k\sigma) \leq ...
6
votes
1answer
51 views
Probability of Heads in a coin
I was wondering, if you flip a fair coin $5$ times, whether you can calculate the probability of getting at least one head is calculated like this:
You can do the complement of getting at least one ...
0
votes
1answer
26 views
uniformly distributed random variable
Suppose that a uniformly distributed random variable X can have each of the seven values
-1, 0, 1, 2, 3, 4, 5
and $Y = X^2-2*X$
I want to get the the probability mass function of Y
I used the ...
2
votes
1answer
12 views
Probability involving unique group combinations
If I have 30 objects and 5 buckets that each hold 6 objects, how many times could I put the objects into the buckets without an object being in the bucket with an object it has previously been grouped ...
1
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1answer
24 views
Conditional Probability of Jointly Distributed Random Variables.
The random variables X and Y are jointly distributed according to the pdf $f_{X,Y} (x,y)=x+y, 0 <x<1, 0<y<1$. Find $P(Y>1/3|X=1/2)$
Since this is a conditional probability, can't I ...
1
vote
1answer
81 views
A probabilistic method
I am trying to study for a exam and i found a assignmet, that i cant solve.
Consider a board of $n$ x $n$ cells, where $n = 2k, k≥2$. Each of the numbers from $S = \{1,...,\frac{n^2}{2}\}$ is written ...
1
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1answer
60 views
Probabilistic interpretation of $\sum_{n \geq 0}\mathrm P_{n}(t)= 1$
The following is a problem from Spiegel's Applied Differential Equations:
The probability $\mathrm{P}_{\mathrm n}(t)$ that a counter (such as a Geiger counter) will register exactly $\mathrm n$ ...
0
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0answers
17 views
Bayesian Inference. how to do marginalization/marginalize out
i am currently trying to implement bayesian in my programming language...
i want to marginalizing my posterior over the nuisance parameters to infer some parameters. However, I dont know how to do ...
2
votes
3answers
74 views
Does a random action have probability?
Imagine I have a coin and I through it to the air and it has side(a) and side(b).Itis absolutely random which side is going to be facing up.
Knowing that,
What probability does the coin have to ...
2
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0answers
59 views
Notationally, what is the difference between $\Pr(X = x)$ and $P(X = x)$? When should I use each?
I'm talking specifically about probability theory. I was reading some stuff about probabilistic graphical models, and they kept switching the notation in this book, but I couldn't discern the ...
0
votes
2answers
28 views
Probability rules with two conditionals only
I have an homework with these two conditions :
P(B|A) = 0.87
P(¬B|¬A) = 0.98
And I have to find P(B). ....I have no idea how to procede from there.
Both events A, and B are dependent and not ...
4
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4answers
160 views
Combinatorics : Which side is heavier?
n coins are given, among which exactly 3 are bad and heavier than the good ones. A balance is used to identify the bad coins. Assume k coins are picked in both sides of the balance at a time. What is ...
6
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1answer
148 views
When is the sum of first $n$ numbers equal to the sum of the next $k$ numbers?
When is the sum $1+2+\cdots + n = (n+1) + (n+2) + \cdots +(n+k)$?
The easiest solution $(n,k)$ is $(2,1)$. For example, $1+2 = 3$. Do any others exist?
Roots of $(n+k)^2 + (n+k) = 2n^2 +2n$ give ...
0
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0answers
25 views
Expectation of operators
Let us define the operator $\vec{L_{op}}=\vec{r_{op}}\times \vec{p_{op}}$ where $\vec{r_{op}}=(x,y,z)$ such that $r_{op,1}f=xf$ and $\vec{p_{op}}=({\partial \over \partial x}, {\partial \over \partial ...
1
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0answers
24 views
naive bayes, understanding the correctness of a model and computation
I implemented naive bayes algorithm to predict an emotion ( happy , sad ) for blogs using the formula provided by Manning's Information Retrieval book
http://nlp.stanford.edu/IR-book/pdf/13bayes.pdf
...
1
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1answer
62 views
(Game Theory) Incomplete Information extension of the 'Centipede' Game
This question is an extension of the 'centipede' game, seen here: Centipede Game.
My prof. posed this to me in class and I can't figure out how to approach this problem..
Imagine in this game, ...
1
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0answers
72 views
+50
Variance of the product of a random matrix and a random vector
If $X$ and $Y$ are independent random variables, then the variance
of the product $XY$ is given by
$
\mathbb{V}\left(XY\right)=\left\{ \mathbb{E}\left(X\right)\right\} ...
1
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2answers
29 views
probability calculation,system downtime,minimum downtime duration
here's the question:
assuming that the system downtime is normally distributed with mean (μ)4.47 sec and standard deviation(σ) of 0.38 sec.
By using the cumulative (to the left) Z score table,
a) ...
5
votes
2answers
80 views
Probability and permutations
I performed an experiment in which an individual had to order 5 items (i.e. his "response" was something like $(3,2,1,4,5)$ or some other permutation). The correct ordering was $(1,2,3,4,5)$ and I ...
2
votes
1answer
28 views
Approximating the logarithm of a Laplace transform
Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$.
Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e.
...
0
votes
1answer
91 views
Using Chebyshev's inequality to obtain lower bounds
I need help with a question I found in Master Stats. I'm unaware of Chebyshev's inequality hence I can't do this question, can anyone help.
Q) A company produces planks whose length is a random ...
