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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7 views

Return probability in markov chain

Given the following markov chain : $p_{0,1}=1$ (if we are in state 0, we must go to state 1) $p_{i,i+1}=p_{i,i-1}=0.5$ There are infinite (countable) states. I defined $T=inf\{n>0 : X_n=0 | X_0 ...
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0answers
15 views

How calculate the probability that there is row in which there is no silver coin?

Data is $n ^ 2$ coins, including $n$ silver. It shall be set at random in $n$ rows, and each row is the $n$ coins. How calculate the probability that there is row in which there is no silver coin?
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1answer
13 views

Reason behind convergence in probability definition

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$ $$\lim_{n\to\infty}\Pr\big(|X_n-X| > \epsilon\big) = 0$$ But ...
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0answers
11 views

Random variable and distribution - number of tests a teacher has to make

$100$ students do a test. The probability of failing the test is $0.6$, those that failed, do a retest, the probability of failing the retest is $0.5$. Those that fail the retest do another retest. ...
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0answers
6 views

pdf for the sum of squared iid normal random variables

I am trying to find the distribution/pdf for the sum of squared $X_i$ iid observations from the normal distribution $X_1 ,..., X_n$ ~ $N(\alpha , \alpha)$, i.e. $X_1 ^2 + X_2 ^2 +...+ X_n ^2$. I was ...
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0answers
7 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
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1answer
17 views

Combinatorics/Probability unordered lists

I don't really understand these unordered lists problems such as... Q: John goes to a store and buys 10 pieces of fruit from the selection of apples, bananas,peaches and pears at random. What is the ...
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1answer
27 views

Expectations of squared sum question

I can't seem to figure out why these expectations turn out the way they do, I am currently studying about the Fisher Information. If $X_1,X_2,...,X_n $ are all iid Poission($\lambda$) , then going ...
3
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0answers
14 views

On the probability of singular matrices containing whole numbers

Today in class - my teacher was teaching determinants . He gave us problems to solve of various kinds , including various row - column operations and determinants properties. But one thing that ...
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2answers
26 views

Probabilty exam question

I would like some help with what direction to take in this question.I find it difficult to decide what rule I need to use when I read a question. Cars pass at an average rate of 1 every 10 seconds. ...
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0answers
23 views

Drawing 6 balls of different colours

Hi I have an exam on Monday and am doing a few probability questions. I have checked the mark scheme for the answer to the following question however the method isn't stated. Could someone please ...
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1answer
42 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
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1answer
75 views
+50

Result of a $2D$ random walk with position dependent probabilities of step

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk:- A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
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0answers
177 views
+100

A fun card game involving probability, getting all 13 ranks (any suit(s)) vs. 5 in a row of red or black.

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, shuffled well before each hand is drawn, and randomly draw cards from it one a time ...
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2answers
60 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
39 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 ...
2
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1answer
22 views

Conditional Probability of Sinking Ship Question

Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a ...
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2answers
19 views

Prove that markov chain is recurrent

I have the following markov chain : $S=\{0,1,2,3\}$ $p_{i,0} = q$ (if we are in one of the states $0,1,2,3$ we can return to $0$ with probability $q$) $p_{i,i+1} = 1-q , i\in\{0,1,2\}$ (if we are ...
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0answers
15 views

Compare 2014 to 1998, 2014 has a 90% chance of being warmer than 1998?

According to NASA, 2014 has a 38% chance of being the warmest year, 1998 has only a 4% chance of being the warmest year. 2014 or 1998 have a 42% chance of being the warmest year. Since I eliminated ...
2
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1answer
30 views

Average difference between two odd numbers of equal length

If I select two different odd numbers of binary length $l$, what is the formula that will tell me the average difference between those two numbers? Note that the high order digit must always be $1$, ...
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2answers
42 views

How to decide the randomness of a dataset by checking the prime numbers inside it?

So it is weekend! I am reading currently a book where I found this sentence: "71 percent of men said they had a 'good sense of direction'. Only 47 percent of women reported same thing.", and I thought ...
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0answers
9 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
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2answers
41 views

Conditional Probability Question. [on hold]

A letter is known to have come from either 'TATANAGAR' or 'CALCUTTA'. On the envelop just two letters 'TA' are visible. What is the probability that the letter has come from (i) TATANAGAR (ii) ...
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2answers
32 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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1answer
22 views

Probability Distribution sampling problem

$\text{*The below problem was asked in geometric distribution section}$ In a population there are $50\%$ Male and $50\%$ Female What is the probability to find $2$ Females in a row out of $10$ ...
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0answers
21 views

Bayes theorem example

For the first question I get a very small probability 3.55%. I use bayes theorem to calculate it, is it correct? It seems a bit small to me. Medical testing Let us imagine that it is discovered that ...
4
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1answer
57 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
-1
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0answers
26 views

Inequality with poisson r.v. [on hold]

Let $r>0$ and $X \sim Poisson(\lambda)$. Prove that ( $e=2.71...$) $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ I can show it for $r \in \mathbb{N}$ by writing expected value as series, ...
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0answers
30 views

$2014$ has about a $90\%$ chance of being warmer than $1998$?

According to NASA, $2014$ has a $38\%$ chance of being the hottest year, $2010$ has a $23\%$ chance of being the hottest year, $2005$ has a $17\%$ chance of being the hottest year, and $1998$ has ...
2
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1answer
33 views

Sum of two normal numbers need not be a normal one

Using the translation invariance of Lebesgue measure how to show that sum and difference of two normal numbers need not be normal ? Normal number in $(0,1]$ is a number $\omega$ such that $\lim_{n ...
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0answers
7 views

Kernel estimate in boundary point

Good moorning, I wonder how to prove that if $X_{1}, \ldots, X_{n}$ are iid from exponential distribution with expected value 1, then the expected value of its kernel density estimator in zero is ...
0
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1answer
12 views

Finding binomial probability, bernoulli trials

The following table lists World Series Lengths for the fifty years from $1926$ to $1975$. Test at the $0.10$ level whether these data are compatible with the model that each World Series game is an ...
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0answers
14 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 : ...
1
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1answer
23 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 ...
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0answers
11 views

Network theory_probability [on hold]

Please help me to understand the probablity i^' sdirect contact N_i (g)={j≠i │ij∈g},of size n_i (g).The size of g is n(g)=∑_(i∈N)▒(n_i (g))/2. Players loose their ob with beakdown probability ...
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0answers
15 views

Ant walking on a coordinate plane

I'm going to raise the difficulty of the original question one dimension, so maybe a refresher will be good... Link: http://puzzling.stackexchange.com/questions/10839/ant-walking-on-a-number-line I ...
1
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1answer
184 views

probability mass function vs. cumulative distribution function

One can understand if probability mass function is known then the cumulative distribution function is known and vice-verse. Can someone tell me how they are related to each other?
0
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1answer
18 views

Throwing a fair die 5 times

You throw a fair die 5 times. What is the probability that the minimum of thrown numbers is 3? I would have said that all possibilities are $6^5$ and that I have $(1*4^4)*5$ ways to get a minimum of ...
1
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1answer
20 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
votes
2answers
38 views

Help me find $P(A \cup B')$ 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). There was an error please try ...
0
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0answers
23 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 ...
0
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1answer
18 views

$\Pr(X+Y\geq1)$

Two random variables X and Y have the following joint pdf: $$f_{X,Y}(x,y)\begin{cases}10x^{2}y & 0<x<1,0<y<x\\0 & \text{otherwise}\end{cases}$$ I am asked to find the marginal pdf ...
2
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1answer
34 views

What are the odds that two people are friends in a network of 20 people?

If person $A$ has 10 friends and person $B$ has 5 friends, and they are in a network of 20 people, what are the odds that persons $A$ and $B$ are friends? I first thought to divide into cases ...
2
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2answers
34 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...
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2answers
303 views

Probability Dealing with Cards

Here is the story. You are dealt 8 cards from a well shuffled deck of 52 cards. What is the probability of getting 4 queens and 4 kings? I understand the probability is quite small but how do we set ...
0
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2answers
1k views

What is the expected number of times a 6 appears when a fair die is rolled 10 times?

Ok, so I think I have a working solution to this problem. Heres how I would solve it: so you look at a 6 appearing as a success and everything else as a failure. So from here you can you use the ...
2
votes
2answers
50 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- ...
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2answers
36 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 ...
4
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1answer
43 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 ...
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0answers
18 views

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

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 ...