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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0answers
69 views

Optimal allocation in network

Given a network (N,g). We want to analyse specializaton matters. Nodes are individuals, and they can product goods and services just like in our usual economy. Individuals can be consumers too. This ...
0
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1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
2
votes
1answer
21 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
0
votes
0answers
17 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
1
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0answers
35 views

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
1
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0answers
16 views

Distribution of sample skewness and kurtosis

I am working on my thesis right now and I'm almost done with it, but just on the last step I encountered some problems with a proof. I have an independent sample $X_{1}, ..., X_{n}$ that follows the ...
4
votes
6answers
58 views

Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables

Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$. I've tried in this way: $$ \begin{eqnarray} P(X < Y) ...
-1
votes
1answer
40 views

Largest value of an expected value? [on hold]

I have $m$ balls and $n$ bins, and I want to find the expected number of non-empty bins. In order to make the problem easier, I decided to find the expected value of empty bins instead, but now im a ...
1
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0answers
24 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
0
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2answers
33 views

A question on probability of choosing coins

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3 \over 4$ and tails with probability $1 \over 4$. Three coins are chosen from ...
1
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1answer
18 views

How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
3
votes
2answers
88 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
3
votes
1answer
25 views

Median of waiting time for $k$-th ace from bridge cards

I can't figure out how to get median of a waiting time from the exercise 36 from W. Feller's book An Introduction to Probability Theory and Its Applications Vol.1 (bold in the quote): ...
3
votes
1answer
25 views

Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi _{2n}^2$.

The Statement of the Problem: Suppose that $X_1,\ldots, X_n$ is a random sample from the $U(0,1)$ distribution and $$ Y_i = -2\log X_i. $$ Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi ...
0
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1answer
22 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + ...
0
votes
1answer
25 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
0
votes
1answer
29 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
0
votes
1answer
30 views

An application of Jensen's Inequality for dependent random variables

Consider dependent and positive valued random variables $A,B$ and $X$. I want to prove that \begin{equation} E[X^2 A] E[B] \ge E[X A] E[X B]. \end{equation} If $A$ and $B$ were scalars, above would ...
0
votes
2answers
22 views

Show that the conditional pmf of $X_i$, given $T = t$, is $Binomial(t, \lambda_i/\lambda).$

The Statement of the Problem: The random variables $X_1, ..., X_n$ are independent and $X_i \sim Poisson(\lambda _i), i = 1, ..., n$. Set $$ T = \sum_{i=1}^n X_i \qquad \text{and} \qquad \lambda = ...
2
votes
2answers
87 views

Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
0
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0answers
24 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
0
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1answer
29 views

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board, given that two colorings are considered the same if one is a rotation of the other?
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3answers
30 views

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many 5 card hands are there that include exactly one pair of two cards that have the same numeric value?
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1answer
41 views

Probability of two heads in a sequence of coin flips [on hold]

A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction.
2
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1answer
26 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
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0answers
26 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show ...
0
votes
2answers
31 views

Central Limit Theorem and understanding mean for a single object

The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students ...
-1
votes
3answers
50 views

Probabilistic problem with balls in a Box. [on hold]

A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white? Please help me ASAP!!!! Thank you!
0
votes
0answers
16 views

Struggling to understand multi-class logistic regression

It is well defined that given a data set of $N$ $i.i.d$ observations $\mathbf{X} = \{\vec{\mathbf{x}}_1, \dots, \vec{\mathbf{x}}_n\}$, along with corresponding target values $\vec{\mathbf{t}} = {t_1, ...
-1
votes
1answer
10 views

probability of predicting positions in a league [on hold]

There are 20 teams in the premier league.how do I work out the probability of predicting the exact position of each team in the league
0
votes
1answer
21 views

Type I error in Normal distributions

Let $X_1,\dots , X_n \stackrel{iid}{\sim} N(\mu, \sigma^2 = 4)$ Test $H_0: \mu = 10$ vs $H_1: \mu > 10$ take a random sample of $n=16$ and reject $H_0$ if $\bar{x}>14$ Find $\alpha$ the type I ...
-1
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0answers
45 views

probability distribution of Complex Gaussian column vector and conditional probability of complex Gaussian column vector

I have column vector $\vec r=[r_1\ r_2]^T$. $$\vec r =hA\vec s +\vec n$$ where $h$ is a complex number , $\vec n \sim \mathcal{C} \mathcal{N}(\begin{bmatrix} 0 \\ 0 ...
0
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1answer
37 views

Type I and type II errors

Let $X \sim uniform(0,\theta)$ we are testing $H_0: \theta = 1$ vs $H_1: \theta >1$ If we know that we reject $H_0$ if $X>0.9$ (1) find $\alpha$, the type I error (2)Suppose that $\theta=1.1$. ...
-3
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0answers
20 views

doubt in programming mathematics [on hold]

anyone pls help me...how this 0.5625 probability comes from in between 5 and 3
-1
votes
1answer
32 views

Given circles with radius R1,R2(has same centre) what is probability of point which lies between R1 and R2 is closer to R2 than R1 [on hold]

There are two circles that have the same center and are of radii $ R_1 $ and $ R_2$.We should place a point in the area formed between $R_1$ and $R_2$. What is the probability that the point is closer ...
0
votes
1answer
26 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
2
votes
1answer
60 views

Probability of getting the same vector result

This is part of a mathematical puzzle I was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
1
vote
2answers
59 views

Expected number of coin tosses [on hold]

Consider a perfect coin with two sides (A and B) each equiprobable $P(A)=P(B)=P(\bar{A})=\frac{1}{2}$. What's the average number of tosses to get (AA) and (AB)? (all tosses independent) Closed form ...
-4
votes
1answer
37 views

Probability of atleast 2 people having a common birthday in a room of 32 people [on hold]

In a room full of 32 people, what is the probability that at least two of them have the same birthday. I considered the different cases like this, 1: No two people have birthday on the same day 2: ...
2
votes
0answers
35 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in ...
1
vote
4answers
151 views

How to understand $E(XY)$ intuitively

I have no trouble understanding $\displaystyle E(X)=\int xf(x)\,dx $ and $\displaystyle E(Y)=\int y f(y)\,dy$ As each $x$ multiplies the corresponding $f(x)$ and we take the integral of it to ...
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0answers
16 views

Aerospace engineering Probability book (graduate)

I am a probability student interested in learning the applications of stochastic differential equations and processes for aerospace problems. I am new to engineering but I can pick up the basic ideas. ...
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0answers
25 views

Probability of multiple wins in a series of evenly matched teams

24 evenly matched players are divided into 6 teams. (Evenly matched by virtue of golf handicaps). Team assignment is random. One player frequently wins, irrespective of which team he is assigned. I ...
1
vote
2answers
29 views

Confusion regarding Burke's theorem

Arrivals occur at rate $\lambda$ according to a Poisson process the service time have an exponential distribution with parameter $1/\mu$ in an M/M/1 queue, where $\mu$ is the mean service rate where ...
1
vote
0answers
34 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
0
votes
1answer
25 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
1
vote
0answers
30 views

A finite field subset sum count

Given $d\in\Bbb N$, pick $N=2^{2d}$ distinct $a_j$ from $\big\{1,\dots,2^{d^2}-1\big\}$ and pick $i$ from $\big\{3,\dots,2^{d}\big\}$. On average how many of $i$-subsets in ...
0
votes
2answers
17 views

Question about finding a distribution without taking into account previous events

We have 8 prisoners, each has a probability of escaping (independently) each day of $0.4$, what is the distribution of the amount of escaping prisoners on the third day? This is the answer: the ...
1
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2answers
19 views

probability of rolling a die 15

if you roll a die 15 times, what's the prob that there are four 6's? Answer is $\binom{15}{4} * (1/6)^4 * (5/6)^{11}$ I am assuming the $(1/6)^4$ comes from the probability you get four 6's, and ...
2
votes
1answer
33 views

Coin flipping game with stop-loss

You play 100 rounds of a coin flipping game where you win \$2 for a head and lose \$1 for a tail on each round. Clearly since the coin tosses are independent the expected winnings are \$50. Now, ...