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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
11 views

Finding a probability density function of a function of three dependent random variables

I have three random variables that are functions of another three random variables by pairs, say: $U=fc(X,Y)$, $V=fc(Y,Z)$ and $W=fc(X,Z)$, with $X$, $Y$ and $Z$ being independent random variables ...
0
votes
0answers
15 views

Maximum of a Gaussian random walk with non-identical steps

Consider a sequence of independent normal random variable $X_1,...,X_n$ with (negative) means $\mu_1,...,\mu_n$ and standard deviation $\sigma_1,...,\sigma_n$. Define \begin{equation} S_k = ...
2
votes
1answer
26 views

In how many ways you can put n white balls and 2n black balls into n boxes if at least one black ball have to be in each box

n - number of white balls 2n - number of black balls In how many ways you can put it into n boxes? It have to be at least one black ball in each box. My idea: First of all let's put one black ball ...
1
vote
3answers
46 views

Probability of word LOLLIES

Consider this probability question. "A four letter 'word' is chosen randomly from the letters of the word LOLLIES. What is the probability of this word containing exactly one L?" Now computing the ...
1
vote
1answer
18 views

Positive information counterexample

An event F is said to carry positive information about an event E ($F\to E$) if $P[E|F]\ge P[E]$ Prove or give counterexample to the following assertion: if $F\to E$ then $E\to F$ My attempt: I ...
0
votes
0answers
25 views

expectation lognormal and normal

I have two random variables $X\sim N(m_{X},\sigma^2_{X})$ and $Y\sim N(m_{Y},\sigma^2_{Y})$ both normally distributed and they're jointly normally distributed as well with correlation $\rho$. I am now ...
0
votes
1answer
25 views

Inequalities for the probability of union and intersection of events

Prove that $\min(1, P(A)+P(B))\ge P(A\cup B)$ $\min(P(A),P(B))\ge P(A\cap B)\ge \max(0,P(A)+P(B)-1)$ Where $A$ and $B$ are events. I don't know how to prove them; Can you give me a hand please?, ...
0
votes
0answers
33 views

Proof of Conditional Probability

Given a set of data points D and mean u which is continous - How to prove the fact below p(x=1|D) = Integral( p(x=1|u)*p(u|D) du ) given that x can take x=1 ...
-1
votes
0answers
25 views

Impossibility and Mode of convergence [on hold]

Let $X$ be a random variable taking values in $[0,\infty)$ and $a$ be a constant. I have that $Pr(X<a)=0$. Is it possible to have $a<0$? If this is true, what is type of convergence?
1
vote
0answers
8 views

Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
0
votes
1answer
30 views

Integrating probabilities

My following problem is of general nature, here is an example to illustrate it. For example let $\left(\xi_i\right)_{i \geq 1}$ be independent and identically Exp(1) distributed random variables. We ...
1
vote
2answers
20 views

Expected Value of Identically distributed random variables

I have a very quick question regarding the expected value of two random variables $X,Y$ that are identically distributed and not necessarly independent. Is this equation valid? $E[XY]=E[X^2]$ If ...
1
vote
1answer
17 views

Positive information of an event proof

An event F is said to carry positive information about an event E ($F\to E$) if $P[E|F]\ge P[E]$ Prove or give counterexamples to the following assertions: 1)if $F\to E$ and $E\to G$, then $F\to G$ ...
1
vote
1answer
32 views

Normal Distribution Worded Problem

Standard deviation = 2.5 mL 98% of bottles must be between 998 mL and 1000mL Pr( 998 < x < 1000) = 0.98 This is a technology exam question, therefore to find the mean I used the method: ...
0
votes
0answers
15 views

What does a presentation on block design and Latin squares consist of?

I read the wikipedia pages of both and I just cannot understand these two concepts. I have a presentation on both of these topics next week and I need some headway on both of these topics.
0
votes
1answer
27 views

Distribution of sample median for a discrete random variable

Say I have a set $S = \{x_1, \dots, x_m\}$, where $1 \le x_i \le n$, all distinct, with median $M$. I take a sample $T$ of size $t$ from $S$, with replacement. I call $Y$ the median of $T$. What is ...
0
votes
0answers
39 views

Upper bound on the covariance of two gamma processes?

Given two binary gamma processes, $X = \Gamma(t; \gamma_1, \lambda_1)$ and $Y = \Gamma(t; \gamma_2, \lambda_2)$, what is their maximum covariance? Applying this answer, it would seem that it is the ...
0
votes
1answer
18 views

Prove that mean square error equals expected conditional variance

I'm a first year grad student in Statistics. The book I'm using mentioned conditional variance, and I wanted to read up more about it. I dove down the google rabbit hole and found this website. I read ...
0
votes
1answer
16 views

Probability of Birth Process [on hold]

Suppose a simple birth process with birth rate $\beta$ starts with two individuals. What is the probabilities that at time $t$ the population contains two individuals?
3
votes
4answers
65 views

When does $P(A|B) = P(B|A)$?

If A and B are events, when does $P(A|B) = P(B|A)$? If it is not always true, please provide a counter example as I cannot.
0
votes
1answer
27 views

Probability question about the chances of cancer [on hold]

Approximately 1 in 14 men over the age of 50 has prostate cancer. The level of prostate specific antigen (PSA) is used as a preliminary screening test for prostate cancer. 7 % of men with prostate ...
3
votes
2answers
44 views

Alternative Monty Hall Problem

So the typical set up for Monty Hall problem, I there are 3 doors where 2 have goats and 1 has a car. I, the contestant, get to randomly guess a door looking to get the one with the car, after this ...
1
vote
0answers
25 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
2
votes
1answer
33 views

More computationally optimal way to solve probability of N or more empty buckets given B buckets and A balls

Problem What is the probability of observing N or more empty buckets given B buckets and A balls, if you throw the balls into any of the buckets with equal probability. Python Simulation ...
0
votes
2answers
19 views

Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $ \theta$ and variance $ a>0$. Let $ Y $ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
0
votes
0answers
23 views

Bounding the size of consecutive sums of independent Bernoullis

Let $\{X_i\}$ be a sequence of independent Bernoulli random variables that take the values 1 and 0 each with probability 1/2. Is the following statement true? For any $\epsilon > 0$, there exists ...
1
vote
1answer
33 views

Meaning of $P(Y|X=x)$

Suppose that $X$ and $Y$ are two random variables on $(\Omega, \mathcal H, P)$ with values in $(\mathbb R,\mathcal B_{\mathbb R})$. I want to understand what is "formally" the expression $P(Y|X=x)$ ...
-2
votes
3answers
41 views

Probability of $X < Y$ [on hold]

Two independent random variables $X$ and $Y$ have distribution functions $\lambda_1e^{-\lambda_1x}$ and $\lambda_2e^{-\lambda_2x}$ respectively. $Pr(X < Y)$ = ?
1
vote
1answer
32 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
0
votes
1answer
22 views

Conditional probability and bayes theorem problem involving a medical test

I have a test that checks if a patient is sick (E = {patient is sick}) and gives either a positive (A={result is positive}) or a negative result. Given that $P(A|E) = 0.95 = P(A^c | E^c)$ and $P(E) = ...
1
vote
2answers
19 views

Probability of observing a false correlation and confidence limits

In oil and gas exploration/development it is common to use acustic impedance derived from reflection seismic surveys to predict the porosity measured in wells drilled in the reservoir. I often use ...
2
votes
2answers
63 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
0
votes
0answers
13 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
0
votes
0answers
19 views

Game theory: Bidding strategy during an auction in a card game

I'm trying to create a mathematical model for the auction process in a card game called Pitch. The specific question I'm interested in solving is: Let $p_i$ represent the probability of a specific ...
3
votes
2answers
67 views

How many ways can $10$ digits be written down so that no even digit is in its original position

If I have the numbers $0,1,2,3,4,5,6,7,8,9$ written down in that order, how many ways can the $10$ digits be written down so that no even digit is in its original position? It would seem that I can ...
0
votes
1answer
23 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
0
votes
1answer
33 views

Probability of the intersection

We know that the probability of the intersection of two independent events is equal to the product of their probabilities? Are there any conditions under which the probability of the intersection of ...
0
votes
3answers
22 views

Median Value + Mode for Hybrid Functions of a Continuous Probability Density Function

To find the median: should I set the integral to 0.5.... but because there are two functions that are non-zero, I am unaware of a method to find the median. To find the mode: would I need to ...
0
votes
2answers
39 views

The probability of $n$ being a square, given the units-digit in its decimal representation

Given a natural number $n\in[1,N]$, the probability of $n$ being a perfect square is $\frac{1}{\sqrt{N}}$. What would be the probability, if we knew the units-digit in the decimal representation of ...
0
votes
1answer
41 views

Is it possible to study information theory while studying a first course on probability?

I'm currently taking a course on intro to probability. The course is not mathematically rigorous and does not invoke theorems from real analysis, etc. The course covers all the way from basic ...
1
vote
1answer
103 views

How to convert a problem to a stars and bars problem?

Continued question from here. With certain questions I have $x_i$ being constrained by various different inequalities, I want to know how to remove these from the problem, to bring me back to a ...
2
votes
3answers
57 views

probability rolling a dice 5 times

I can't solve this problem: What is the probability that, when rolling a dice 5 times, the number of times when you get a 1 or 2 is greater than the number of times when you get a 6. any help?
1
vote
1answer
44 views

Probability of Grouping boys and girls

10 boys and 2 girls are divided into 3 groups of 4 each. The probability that the girls will be in different groups is?
1
vote
0answers
35 views

Conditional Probability with Normal Distributions

Let's say that I have $3$ random normal variables, $A$, $B$ and $C$. They all have a standard deviation of $17.526$, while $A$ has a mean of $143$, $B$ of $139$, and $C$ of $129$. I want to ...
0
votes
1answer
32 views

Probability of order statistics with numerous conditions

Let $Y_i$ be the i-th order statistic of a continuous random variable $Y$ and let $z_{k-1}\leq z_k$ for all $k$. Let $1\leq j\leq n-1$. How can I evaluate or rewrite $$Pr(Y_{n-j-1}\leq ...
1
vote
2answers
36 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
1
vote
0answers
47 views

Expected frequency of most frequent die roll

Suppose we have an fair $m$-sided die, and we roll it $n$ times. What is the expected frequency $E(n, m)$ of the most frequently rolled face? If we fix $n$ we can calculate $E(n,m)$ like so. Let ...
3
votes
5answers
148 views

Probability in a knock-out tournament

Maths newbie so please go gently. Imagine if you would: 4 teams in the semi-final of a soccer tournament A,B,C,D. A,B and C all have 20% chance of winning the tournament. D however, is the favourite ...
2
votes
2answers
37 views

Collision of 8 Digit, Base-36 Numbers

I have an algorithm that generates a random 8 digit, base 36 number with uniform distribution. Thus, this algorithm can generate $36^8$ unique numbers. I run my algorithm 10,000 times, and write ...
1
vote
2answers
16 views

Odds of drawing the same ball in consecutive rounds

Consider a game where $k$ out of $N$ numbered, distinct balls with labels $1$, $2$, ..., $N$. How high is the probability that one or more numbers are repeatedly drawn in two consecutive rounds? More ...