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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Uniformly Distributed random varibles

Question:Suppose X is a uniformly distributed random variable with possible values 1,2, ..., 10. Compute the expected value and variance of X. I have started with making a column (x on the left and y ...
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1answer
16 views

Find the 95% confidence interval and interpret the results [on hold]

A random sample of 38 200-meter swims has a mean of 3.96 minutes and the population standard deviation is 0.06 minutes. Construct a 95% confidence interval for the population mean time. Interpret the ...
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6 views

Using an “auxiliary random experiment” to achieve a desired significance level

My question is somewhat simple, but, nonetheless I am not entirely convinced I am solving it correctly. I need to use the use the Neyman Pearson Lemma to test for $H_o : \theta = .5$ vs. $H_1 : ...
1
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1answer
13 views

Expected value for sum of iid normal variables squared

Let $X_i$ be iid from a $N(\alpha, \alpha)$ distribution. I am trying to find $E[\sum_1^n X_i ^2]$ and thought that I would be able to transform the statistic $\sum_1^n X_i ^2$ into a chi-squared ...
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0answers
19 views

Joint Probability Question

I have a question regarding join distributions. For this question, I have to find the probability that P(X+Y=0). I've attempted multiple different ways to solve this problem, but I keep getting 0 as ...
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1answer
12 views

Covariance and Correlation in Multinormal random variable

Find the covariance and correlation of $N_i$ and $N_j$, where $N_1, N_2, \ldots,N_r$ are multinormal random variable. At the beginning, I think that I have: ...
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0answers
6 views

Lists of common sufficient statistics

Can someone suggest a source for common sufficient statistics for exponential families? For example, I'm looking for something more comprehensive than the Wikipedia page for sufficient statistics, ...
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0answers
10 views

Rationale behind formula relating death probability and observed mortality rate

With $M_{x,t}$ stands for the time-$t$ observed mortality rate between ages $x$ and $x+1$ (formulas given below) and $q_{x,t}$ the time-$t$ death probability between ages $x$ and $x+1$ (the ...
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0answers
12 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 ...
-1
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0answers
33 views

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

There are $n ^ 2$ coins, and $n$ of them are made of silver. The coins are set at random in $n$ rows, with each row having $n$ coins. How do we calculate the probability that there is a row in which ...
1
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1answer
19 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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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
10 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
10 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 ...
2
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1answer
21 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 ...
3
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0answers
15 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
27 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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0answers
16 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
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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0answers
10 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

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
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 ...
0
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1answer
28 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 ...
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0answers
22 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 ...
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2answers
44 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) ...
0
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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 $10$ ...
2
votes
1answer
31 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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0answers
31 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 ...
0
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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 ...
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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 ...
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
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 ...
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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 ...
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1answer
13 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 ...
0
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1answer
19 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 ...
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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 ...
-2
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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 ...
4
votes
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 ...
2
votes
1answer
37 views

How many numbers must be selected from 100…999 so that three of them have the same sum of digits? [on hold]

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 sum of the digits in each one. ...
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2answers
15 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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2answers
61 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
14 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
11 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
37 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 ...
1
vote
1answer
22 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
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 ...
-1
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0answers
14 views

Random variables set representation in the sample space [on hold]

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
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
26 views