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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1answer
48 views

Probability of walking into a pole

I have a problem I am thinking about...someone posted on Facebook about randomly walking into a pole and it got me thinking. Say there is a straight line of 100 metres, and in that straight line is ...
0
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1answer
25 views

Is truncating a discrete probability mass function possible?

I have random variable X, and probability distribution: $P[X = A] = .4$ $P[X = B] = .3$ $P[X = C] = .2$ $P[X = D] = .1$ I want to create a conditional probability with event F. Where F is the ...
0
votes
3answers
116 views

Identify the sample points in the events $A \cup B$

A pair of fair dice is tossed. Events $A$ and $B$ are defined as follows : $A$: (The sum of the numbers on the dice is $3$) $B$: (At least one of the dice shows a $2$) Identify the sample points in ...
0
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3answers
361 views

Selecting groups of items: How many ways can we divide n students into groups of two?

I have a question very similar to the post in the link below. But, what do we do when we are given a variable for the total number of students, not a constant number? Here is the modified version of ...
-2
votes
1answer
72 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
0
votes
1answer
116 views

Expected value for Head/Tails

There are $N$ coins placed in a line. A coin may be facing head/tail direction with $0.5$ probability. Now I need to find number of pairs of coins $(i,j)$ such that $i<j$ and on index $i$ , I ...
0
votes
2answers
262 views

Probability of picking balls out of bins

Question: You have two bins with four different balls in each bin. Bin A: 2 White Balls and 2 Black Balls Bin B: 3 Black Balls and 1 White ball You cannot tell which bin contains what balls. Given ...
1
vote
1answer
162 views

Finding the probability of exactly one event in a series of independent events, why used (-1)?

Learning programming and trying to understand this example. Given multiple independent events, each with a probability of occurring, what is the probability of just one event occurring? If we have ...
5
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2answers
127 views

When does pairwise independence imply independence?

We know if a collection of events are independent, then they are pairwise independent. In general, the converse is not true. However, I'm wondering if there's a condition under which the converse ...
5
votes
1answer
193 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
2
votes
1answer
36 views

Matching moments implies matching densities?

If $X$ and $Y$ are random variables with matching moments (ie: $\mu_X^i = \mu_Y^i (\forall i \in \mathbb{Z}^+)$ then are the density functions of $X$ and $Y$ identical (almost everywhere)? Idea: I'm ...
0
votes
2answers
113 views

A fair coin is flipped until the first tail appears, in general we win \$ $2^k $. St. Petersburg problem.

For the St.Peterburg problem (Example 3.5.5), find the expected payoff if (a) the amounts won are $c^k$ instead of $2^k$, where $0 < c < 2$. (b) the amounts won are $\log(2^k)$. The original ...
0
votes
2answers
64 views

CDF of sum of 3 dependent random variables

Given three dependent random variables, $S_1, S_2$ and $S_3$, such that $0 < S_1, S_2, S_3 < \infty$ and assuming known their joint PDF $f_{S_1,S_2,S_3}(s_1,s_2,s_3)$ I would like to find the ...
1
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0answers
41 views

Uniform Convergence and exponential inequality to demonstrate Stirling's formula

If $0<R\leq\sqrt n$ and $Z_{k} \sim exp(1)$ are independent and identically distributed as prove that $$P\left[\left|\dfrac{Z_{1}+Z_{2}+\cdots +Z_{n} -n}{\sqrt n }\right|\leq R\right] \geq 1- \...
0
votes
1answer
27 views

Computing the probability of two variables of the same sample.

Problem: The mean of a variety of apple is 400g with a standard deviation of 50g. If we choose 2 random apples of this variety, what would be the probability that the first one weights 150g more than ...
1
vote
2answers
81 views

Help finding value of N that minimizes a sum

Suppose we have the following inequality: $\sum\limits_{k=N+1}^{1000}\binom{1000}{k}(\frac{1}{2})^{k}(\frac{1}{2})^{1000-k} = \frac{1}{2^{1000}}\sum\limits_{k=N+1}^{1000}\binom{1000}{k} < \frac{1}...
1
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0answers
262 views

Is the new drug superior? Probability & Binomial

A standard drug is known to be effective in 80% of the cases in which it is used. A new drug is tested on 100 patients and found to be effective in 85 cases. Is the new drug superior? (Hint: Evaluate ...
1
vote
1answer
29 views

Calculation of conditional probability

A problem as following: (from Prob, statistics, and random processes for electric engineering, p.264) If I want to find $P(Y\leq y \mid X=+1)$, it can be calculated as following: $P[N+1\leq y]$....
0
votes
1answer
64 views

When computing the CDF from a PDF, why is the integral bound a different variable? $F(x) =\int_{-\infty}^x f(t)\,dt$

Right now, I know the variable in the integral must be x, otherwise, the final result does not match the published CDFs of popular distributions. But I don't know why conceptually this is. If the CDF ...
0
votes
0answers
35 views

Integrability in conditional expectation.

Suppose $X$ is a random variable in $(\Omega,\mathcal F)$.$\mathbb E\left(|X|\right)<\infty$. $Y=\mathbb E[X|\mathcal F_0]$ ,here $\mathcal F_0\subset\mathcal F$. Then I want to show $Y$ is ...
0
votes
1answer
37 views

Conditional Probability with Anti Set

Ethnic Group O A B AB 1 .082 .106 .008 .004 2 .135 .141 .018 .006 3 .215 .200 .065 .020 A given population is in ...
4
votes
4answers
772 views

Gambling problem

Question Robert will win $\$1$ with probability $\frac{1}{4}$, win $\$2$ with probability $\frac{1}{4}$, and lose $\$1$ with probability $\frac{1}{2}$ in a bet. Each bet is independent. Determine ...
0
votes
2answers
144 views

PMF for a multiple choice quiz [closed]

A student takes a multiple choice test of $20$ questions where each question has $5$ possible options for the answers. Suppose the student answers each question at random. Let $R$ be the random ...
1
vote
1answer
118 views

The smallest integer $n$ for a Poisson distribution

Along a stretch of motorway, breakdowns require the summoning of the breakdown services occur with a frequency of 2.4 per day, on average. Assume the breakdowns occur randomly and that they follow a ...
1
vote
0answers
147 views

Does clever noise exist?

This question is about a random noise, which is called clever if its distribution function satisfies a certain condition. Let $Y_0$ and $Y_1$ be two continuous random variables on the same ...
1
vote
1answer
187 views

Problem about number of trials until first success [duplicate]

The problem is like this: A standard deck contains $52$ cards, $4$ each of $2,3,4,5,6,7,8,9,J,Q,K,A$. Now start the following process. Pick a random card from the deck, show it, and then return it to ...
2
votes
0answers
128 views

Finding the limiting probability distribution

I found this problem in Shiryaev's Problems in probability (Problem 3.4.14). Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = \...
7
votes
4answers
309 views

Discover where Bob is sleeping using hidden Markov chains

Bob lives in four different houses $A, B, C$ and $D$ that are connected like the following graph shows: Bob likes to sleep in any of his houses, but they are far apart so he only sleeps in a house ...
5
votes
1answer
2k views

What is the difference between multinomial and categorical distribution?

Both seem to result in one of k different separated outcomes, and Wikipedia says these are often conflated. Despite reading the explanation of the difference on the article about multinomial ...
1
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2answers
48 views

How to attack this probability question directly?

Below is a example probability question I found in my book. I want to attack this question directly instead of using the complement approach. How do I do that? My book always uses the complement ...
0
votes
3answers
31 views

How to compute expected value

How do I solve the expected value of this problem, if I have already calculated the pmf? Let $X$ be a random variable with cumulative distribution function given below: $$F_X(x) = \begin{cases}...
1
vote
0answers
155 views

Distribution of family's disposable income, given pdf, find $F(y)$

Problem: In a certain country, the distribution of a family's disposable income, $Y$, is described by the pdf $$f(y) = ye^{-y}$$ for $y \geq 0$. Find the $F(y)$. Attempt: In the book there it ...
3
votes
1answer
84 views

Simple question about weak law of large number with characteristic function version

I was reading a textbook about showing the following Weak Law of Large Number but I stuck in some intermediate steps. Here is the statement I work with Let $\{X_i\}$ be i.i.d. random variables ...
1
vote
1answer
23 views

Find distribution of a bernoulli funtion of a unifrom random variable?

I have a uniform random variable $\theta \in [-\pi,+\pi]$. I also have a bernoulli function of this random variable $G(\theta)$, defined as follows, \begin{align} \begin{cases} 1 & \text{if $ - ...
0
votes
1answer
33 views

Show that if $X(\omega) = \infty$ then $EX = \infty$

I am trying to show that if $X(\omega) = \infty \space\forall \omega \in A$, $P(A) > 0$ and $X \ge 0$ then $EX = \infty$. The problem comes with a hint: $$EX = E\{X[I(A) + I(A^c)]\} = E[XI(A)] +...
1
vote
3answers
225 views

Probability of the horse winning, given the chance of rain

Here's the question: In the past two racing seasons Seahorse has won 55% of the time if the track is dry. On rainy days when the track is muddy he won only 30% of the time. For the next ...
0
votes
3answers
62 views

Probability and counting cards

The problem goes like this: "I am given 7 cards from a regular 52 playing card deck." "Find the probability that there are at least 3 of the cards equally high (e.g. that there are 3 or more jacks). ...
0
votes
3answers
230 views

Probability with colored flags

A signal has $6$ flags, each flag can be blue, white or red. Possible signals formed is $n^r = 3^6 = 729$ possible signals formed How many different signal can be made from $6$ flags of which $3$ ...
0
votes
3answers
2k views

round table seating probability

There are $6$ people, let's call them - (a,b,c,d,e,f), to sit at a round table. The number of ways they can arrange themselves is $(6-1)! = 5! = 120$ ways. What is the probability that person 'a' ...
0
votes
1answer
35 views

Expression of the thresold with expected degree in a Random Geometric Graph

$n$ points ($P_i$) are distributed uniformly on the surface of an unit radius sphere. 2 points are interconnected if the distance between them is $\le r$ (thresold). We call the degree of point $i$ ($...
0
votes
1answer
59 views

Binomial distribution for a poll

I'm looking for help with this question on the binomial distribution. ...
2
votes
1answer
88 views

Conditional Integral of Square of Brownian Motion?

I am struggling to compute the expectation and variance of the following, where $W(s)$ is a standard Brownian motion: $$ X := \int_{0}^{A}W(s)^2ds$$ $$ Y:= \int_0^AW(s)ds $$ $$E[X\mid Y] = \space ?$$ $...
0
votes
1answer
220 views

Dirichlet distribution, sum of Beta distributions

I currently have a problem about Dirichlet distributed Variables. In one of the papers I am currently reading it says: Let $S=(S_1,...,S_m)\sim Dir(\delta\omega_1,..., \delta \omega_m)$, with $\sum_{...
1
vote
2answers
22 views

Probability of number drawing

The number 1,2,3,4 are written on slips of paper and 2 slips are drawn at random one at a time without replacement. What is the probability the first number is 2 or the sum is 5?
1
vote
1answer
44 views

How do I find $E[X_1|X_1<X_2]$ when $X_1$ and $X_2$ are independent $N(0,1)$ random variables? [closed]

What property of Gaussians do I have to exploit to solve this?
0
votes
0answers
101 views

Binomial Distribution/ Law of large numbers

I currently have the problem to establish that $ \underset{n \rightarrow \infty}{\lim} \sum_{k=0}^{\lfloor n\cdot l\rfloor} \binom{n}{k} (s)^k(1-s)^{n-k} = \begin{cases} 0, & \text{wenn}~l< ...
0
votes
1answer
92 views

A probability problem, Maximum likelihood or Bayesian updating?

I see an interesting probability problem: Suppose you were playing table tennis with A, you did not know the strength of A. Since the start, A had won 3 scores consecutively. What was the ...
0
votes
1answer
26 views

Probability of a Brownian Motion to fall in a bandwidth

Let $X_t$ be defined as $$ X_t = X_0+\int_0^t\sigma_{0}\,dW_s, $$ where $W_s$ is a Wiener process and $\sigma_0\in\mathbb{R}^{+}/{0}$. Which is the probability $$ \mathbb{P}\left[a<X_t-X_0<b\...
2
votes
1answer
102 views

Probability notation question: differences between undergraduate and graduate texts

Suppose $X$ is a random variable. In most undergraduate math texts, one writes the expected value of $X$ as $\text{E}X$ or $\text{E}[X]$. Similarly, the probability that $X$ is greater than some value ...