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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26 views

Conditional probability of overlapping intervals of a Poisson process

I am trying to solve the following problem: Suppose you have a Poisson process with rate $\lambda$. Now define $Y_n =\sum_{i=n}^{n+N-1} X_i$ as the sum of N subsequent inter-arrival times where $X_i$s ...
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11 views

Frequency function for convolution

Suppose A and B are independent discrete random variables and each assumes values 0,1,2 with equal probabilities. Find the frequency function of A+B? Anyone can guide me with all the correct steps ...
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0answers
33 views

Probabilistic proof for sphere covering upper bound

I would like to show an upper bound for the number of $d$-dimensional spheres needed to cover some closed, bounded subset of $\mathbb{R}^d$, like a cube or another sphere. I could do this by placing ...
5
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4answers
742 views

Probability of having at least one pair by drawing 4 shoes from 12 pairs.

There are $12$ pairs of shoes in a cupboard. $4$ are drawn at random. What is the probability that there is at least one pair? My first attempt: If we chose a pair at first and then draw any two at ...
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2answers
60 views

Discrete math probability function rolling 12 sided die

Given a 12 sided fair die write down a probability function that gives the probability of rolling x '7's from 20 rolls. I'm not really sure where to start on this. But I know there are 12^20 total ...
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2answers
94 views

Cards in a 5x5 grid- Probability of a diagonal of all hearts

All the aces, 2’s, 3’s, 4’s, 5’s and 6’s, as well as the jack of diamonds are taken from a regular deck of 52 playing cards, and then placed face up on a table in a 5 × 5 square grid configuration ...
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0answers
11 views

Sum of the $L$ maximum elements from a collection of RVs

Given a collection of $N$ $\chi^2$-distributed RVs (iid, 2 degrees of freedom), $\{X_1,\dots,X_N\}$ what is the distribution of the sum of the $L$ largest of them? That is, what is the distribution ...
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1answer
26 views

Infinite Coin Flips- Set Notation language

Suppose a coin is flipped infinitely many times and we take as our sample space $S$ all possible infinite length sequences of heads and tails. For $n\geq1$, let $A_n$ be the event that in the first ...
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1answer
23 views

Intuition behind precalculus probability..very simple

Two dice are rolled. What is the probability that only one of the dice shows a six? So, the probability that the first dice rolls a six is (1/6) Thus, the probability that the second dice does not ...
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0answers
60 views

UMVUE for $\theta^2$

Let $X_1,...X_n$ be a random sample with distribution $\text{Normal}(\theta,1)$. Find the UMVUE for $\theta^2$ What I´ve done so far: I have already shown that $T=\sum_{i=1}^nX_i$ is a complete ...
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0answers
53 views

Labeled balls and urns. Finding the probability for a specific pairing.

I was reading a text and came across this problem. The wording is a little terse so I'm not clear on why the result is true. Given N balls and N urns, all of which are labeled. The balls are ...
2
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1answer
74 views

Use Bayes's Theorem to Predict Success

I have a group of $n$ events. The successes don't all come in at once, and and I want to try to predict the actual success rate $s$. The number of successes showing in the system at any given time can ...
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1answer
45 views

2 simple statistics questions regarding probability and means.

Fire alarms go off in the engineering building an average of 13 times per year. Find the probability of more than one fire alarm going off in the month of December. For this one, I am uncertain on ...
1
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1answer
76 views

How many ways can 12 different pennies be distributed to four people without each person getting exactly 3?

Question: How many ways can $\mathbf{12}$ different pennies be distributed to four people without each person getting exactly $\mathbf{3}$? My thoughts: I'm really not too sure how to approach this ...
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1answer
20 views

Probability, Mass Function

For what values of constant C does the function $p(k)=C3^{−k}+4^{−k−1}$ defined on the non-negative integers k=0, 1, 2, … constitutes a (probability) mass function? The analogous probability ...
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1answer
228 views

Fine the value of k such that this game is fair (E(X) = 0)

From an ordinary deck of 52 cards, cards are drawn at random and with replacement until three aces are drawn. If it takes more than ten draws to obtain the three aces, the player loses k dollars; ...
2
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1answer
49 views

The expected value of an order statistic for normal random variables

Let $X_1$ and $X_2$ be a random sample from normal distribution with mean equal to zero and variance $\sigma^2$. Prove $E[X_{(1)}]= \frac{-\sigma}{\sqrt{\pi}}$. May I have to standarize the sample? ...
0
votes
1answer
18 views

Conditional gaussians, particular calculation

I'm looking for confirmation that my solution to this problem is correct. The result seems unintuitive. Given $\{X_i\}_{i = 1}^{10}$ $0$ mean jointly gaussian RVs with $\mathbb{E}X_iX_j = 2^{-|i - ...
1
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3answers
58 views

What is the expected value? Three dice are rolled. For a 1 dollar bet you win 1 dollar for each 6 that appears (plus dollar back). No 6, lose dollar.

Three dice are rolled. For a 1 dollar bet you win 1 dollar for each 6 that appears (plus your dollar back). If no 6 appears you lose your dollar. What is your expected value? Practice midterm ...
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2answers
34 views

Discrete Probability Camera Factory

At a camera factory, an inspector painstakingly checks 20 cameras and finds that three of them need adjustment before they can be shipped. Another employee mixes the cameras so up so that no one knows ...
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1answer
38 views

Convolution of two Uniform random variables

We have $X \sim \mathrm{Unif}[0,2]$ and $Y \sim \mathrm{Unif}[3,4]$. The random variables $X,Y$ are independent. We define a random variable $Z = X + Y$ and want to find the PDF of $Z$ using ...
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2answers
79 views

Pascal Triangle Ball Conundrum [closed]

Imagine we have a Pascal Triangle Pin Board: A ball is dropped, and at every pin it has an equal chance of falling left or right. If we drop $32$ balls, what is the probability that the final ...
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2answers
29 views

Given $\{A_n\}$, with $P\left(A_n\right)\to 1$, there exists a subsequence $\{n_k\}_{k\ge 1} \;$ such that $\; P\left(\bigcap_k A_{n_k}\right) > 0$

This is problem 4.21a is Resnick's Probability Path. Suppose $\{A_n\}$ is a sequence of events. If $P(A_n)\to 1$ as $n \to \infty$, prove that there exists a subsequence $\{n_k\}$ tending to ...
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0answers
44 views

Probability question, 6 A people, 14 B people, choose 3 for activity A, 2 activity B, 6 for activity C, what is the number of ways total?

An upcoming super bowl party includes 6 New York Giants fans and 14 Patriots fans. The host will randomly assign guests (choosing names out of a hat) to bring food as follows: 3 to bring wings, 2 to ...
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0answers
11 views

How can positive log-likelihood of an ARIMA fitting be crosschecked by a heuristic technique ?

I use R to fit an ARIMA(1,2,1) model to the observations. In the output, I get positive log-likelihood which is unusual. I want to crosscheck this result. I calculated the probabiliy density function ...
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1answer
39 views

Probability involving the average of a simple random sample

So here's the question: Maria is shopping for wireless routers and is overwhelmed by the number of options. In order to get a feel for the average price, she takes a random sample of $75$ routers. The ...
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0answers
23 views

How to find the probability density function for a function's values?

If I have a function $f(x)$ with a domain $[a,b]$ and range $[c,d]$, how do I find the moments of the values the function attains in the domain? More importantly, how do I find the PDF for the ...
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0answers
13 views

Bayes Net variable elimination

I'm taking a class that involves Bayes Net and can't understand one place where they combine $P(A|B, E)$, $P(j|A)$ and $P(m|A)$ into $P(A, j, m|B, E)$ and then further simplify to $P(j, m|B, E)$, ...
3
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0answers
45 views

Understanding Markov's inequality

Here is a statement of Markov's inequality. Suppose that $X$ is a random variable and that $g: \mathbb{R}\to [0,\infty]$ is Borel measurable and non-decreasing. Then, for any real $c$, $$ ...
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2answers
34 views

Game of Bridge probability

In a game of bridge, choose 13 cards from a deck at random (use equally likely probability). Question: What is the probability you get NO spades? What is the probability you get no card higher than 9 ...
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1answer
40 views

Colored Noise Channel Capacity Derivation in Elements of Information Theory (Cover & Thomas)

On page 277 in Elements of Information Theory, Second Edition by Cover & Thomas the derivation of the information capacity of a colored (Gaussian) noise channel is performed. While the math is ...
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0answers
15 views

Applying Chernoff Bounds with k-wise independence

Ordinarily, Chernoff bounds can be applied to groups of events where they are all mutually independent. However, if these $n$ events are only $k$-wise for arbitrary $k$, is there anything that ...
1
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1answer
43 views

Joint PDF involving independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with parameters $a$ and $b$ respectively. Let $U=\min(X,Y)$ and $$V=\begin{cases}1,& U=X\\ 0,& U=Y.\end{cases}$$ Find the joint ...
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1answer
33 views

Probability that coordinate of a dot within a square less than random parameter Z

From square with vertices (0;0), (0;1), (1;1), (1;0) random dot was taken. It has coordinates (a;b). a and b are inside interval [0;1]. For random parameter z that is between [0;1] find probability ...
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0answers
24 views

When convergence in distribution for $X$ and $Y$ implies for $X+Y$

Let $X_n\rightarrow_L X$ and $Y_m\rightarrow_L Y$. We usually cannot assume that $X_n+Y_m\rightarrow_L X+Y$, since $X_m=^dN(0,1)$, and $Y_m=-X_m$, $X_m+Y_m\rightarrow 0$. So, when can be sure that ...
2
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2answers
24 views

Find the prob. to win third prize in lottery game

I have a question about prob. The question is "A lottery game that chooses 5 numbers out of 50 numbers. What is probability to win the third prize which is the case matching 3 out of the 5 numbers " ...
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0answers
18 views

Constrained sums of conditional multivariate hypergeometric distributions

My first post on math.stackexchange so please let me know if it is a poor one. I am developing a method for making a map using an MCMC algorithm. Each pixel in a map can one of a set of possible land ...
5
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2answers
150 views

Second derivative of $\int_\mathbb{R}\cos(tx)dp(x)$

Let $p$ be a probability on $\mathbb{R}$ and $$f(t):=\int_\mathbb{R}\cos(tx)dp(x).$$ I want to show that if $f''(0)$ exists then $$f''(0)=\lim_{t\to 0}2\frac{f(t)-1}{t^2} \: \:(\star).$$ By ...
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0answers
24 views

Prove that $W_n := (X_n,Y_n)$ is a Markov chain and determine the transition probabilities.

Let $X_n$ be an irreducible, aperiodic, positive recurrent Markov chain $(\lambda,P)$ on a state space $I$, with stationary distribution $\pi$. Let $Y_n$ be Markov$(\pi,P)$, and independent of ...
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1answer
23 views

Probability of cards

I have this question to answer... A card is drawn at random from a pack of 52 playing cards. The card is replaced and a second card is drawn. This card is replaced and a third card is drawn. ...
1
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1answer
60 views

Proof of if two random variables have the same distribution then they have the same moment generating function.

I am trying to prove that if $X$ and $Y$ have the same distribution, then they have the same moment generating function: $M_X(t) = M_Y(t)$ for all $t \in \mathbb{R}$. I came up with a proof, but am ...
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4answers
179 views

How many tickets should I buy to win a prize in lottery? [closed]

Each ticket in a lottery contain a single "hidden" number according to the following scheme: 55% of the tickets contain a 1, 35% contain a 2, and 10% contain a 3. A participant in the lottery wins a ...
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2answers
20 views

Find the probability of binary bit when sent to channel

I have a question about Bayesian rule. My question is For a certain binary communication channel, the prob. that any bit sent is a $0$ is $0.49$. An error occurs with probability $0.08$ given that a ...
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0answers
9 views

Product of log-concave densites

I have a density function $f(a,b,c,d)$ for random variables $A,B,C,D$ which factors as $f(a,b,c,d)=f_{A}(a)f_{B}(b|a)f_{C}(c|b,a)f_{D}(d|a,b,c)$. where $f_{A},f_{B},f_{C},f_{D}$ are log concave in ...
0
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1answer
23 views

If I have that $Z$ is a random variable, and $f$ is a measurable function, how can I show that $E(f(Z)Y |Z) = f(Z)E(Y|Z)$?

I have that $Z$ is a random variable, and that $f$ a measurable function, and would like to show that: $$ E(f(Z)Y |Z) = f(Z)E(Y|Z) $$ This was under wikipedia's expectation page under the "pulling ...
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0answers
28 views

Some sort of binomial distribution

There are $m$ machines which are working. There are $r$ backup machines. If one or more of the working machines break down then we use the backup machines. We try to maintain the number of working ...
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2answers
42 views

If $X$ is a random variable, what is $E(X^{0})$?

I am trying to find what $E(X^{0})$ is, if $X$ is a random variable. My approach is to first find $X^{0}$. I know that for any value $x^0$, if $x \neq 0$, then $x^0 = 1$. However, if we have a random ...
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2answers
69 views

How to prove $Pr\{(A \cap \overline{B})\cup (\overline{A} \cap B)\} = Pr(A) + Pr(B) − 2 Pr(A \cap B)$

I've managed to expand the left side to $$Pr(A \cap \overline{B}) + Pr(\overline{A} \cap B) - Pr((A \cap \overline{B})\cap (\overline{A} \cap B)) $$ and from there to $$ \{Pr(A) * Pr(\overline{B}|A)\} ...
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3answers
610 views

Operator $T \colon L^p \to L^p$ is a conditional expectation

I'm trying to solve this problem: Let $(X,\mathcal{B},\mu)$ a probability space and $T \colon L^p(\mu) \to L^p(\mu)$ a continuous linear operator ($1 \leq p < \infty$ ) with the following ...
2
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2answers
107 views

Very fascinating probability game about maximising greed?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number ...