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)

1
vote
1answer
31 views

Find the CDF for this function

Let $R$ be a continous random variable where the sample space is $-1\leq r\leq 1$ with the following probability density $$f(r) = \begin{cases} \frac{1}{4}& \mbox{for} \quad -1 \leq r \leq 0 \\ ...
0
votes
2answers
30 views

Showing that $\mathbb{E}|X\ln X| < \infty$ and $\mathbb{E}Z = \mathbb{E} X \ln X$ for given PDF of $Z$.

$X$ is real random variable such that $\mathbb{P}(X > 0) = 1$, $\mathbb{E}X^2 < \infty$, $\mathbb{E}X=1$. Let $Z$ be real random variable such that $\mathbb{P}(Z \in ...
2
votes
1answer
31 views

Graphs: probability of two vertices, chosen at random, of being connected by a link

If I choose the vertices h and k in a simple graph uniformly, I know the probability of them being joined by an edge is $ \frac{2e}{n(n-1)} $ , where: e is the number of edges in the graph, n is the ...
1
vote
1answer
18 views

on limits of cumulative density functions

Theorem 1.5.3 of Statistical Inference by Casella and Berger States that the function $F(x)$ is a cdf if and only if the following three conditions hold: 1) $\lim_{x \to - \infty} F(x) = 0 \text{ and ...
1
vote
1answer
19 views

Expected number of draws of a labeled ball knowing it's already been drawn once

Suppose you are drawing labeled balls from an urn with replacement, and you record what you've drawn. You have $n$ balls and draw $m$ times. My question is, for the balls that you've drawn, what is ...
1
vote
3answers
32 views

Probability of caugh at least 1 of one type of fish

In the lake we have got 3 types of fish: k - number of roach 2k - number of crucian 4k - number of perch Mr Smith caught 7 fish. What is a probability that Mr Smith caught at least 1 roach. My ...
1
vote
0answers
52 views

Probability of real roots

Let's take random $a, b \in [0,2] $. What is a probability that equation $ax^2+bx+1=0$ has two real roots? My solution: $\Delta = b^2-4a>0$ $a< \frac{b^2}{4}$ $P( \Omega) = 2 *2=4$ ...
0
votes
1answer
16 views

Random process with stationary independent increments determined by first order distribution?

It says in my random processes book that if a random process $X_t$ with stationary independent increments has value $0$ at the start ($X_0 = 0$) then it is completely determined by it's first order ...
3
votes
1answer
49 views

Guess the smallest number

Three people play a game where each of them writes down a positive integer at the same time. The one who writes a unique and smallest number wins one dollar from every other person. This means if two ...
0
votes
1answer
23 views

Calculation of Conditional Expectation

I have problems with the following exercise: Let $\Omega=[-\frac{1}{3},\frac{1}{3}]$, $\mathcal{F}=\mathcal{B}(\Omega)$ the Borel-$\sigma$-algebra on $\Omega$ and P the Lebesgue-measure. ...
0
votes
1answer
22 views

Hit or Miss Method Question

I´m working on trying to prove a relation when using the Hit or Miss Method. Suppose we have $g$ bounded in $[0,1]$ and suppose $0 \leq g(X) \leq b$ for $0 \leq x \leq 1$. Let $U1, U2$ be independent ...
0
votes
0answers
20 views

Problem about Random walk and Stopping time.

Here is an example in "Probability with Martingales" My questions are: (1)Does equation (a) hold for $T=\infty$? (2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ...
0
votes
1answer
22 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
23 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
30 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
52 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
30 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
26 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
34 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
27 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
10 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
32 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
23 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
18 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
16 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
28 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
51 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
21 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
17 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
66 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
47 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
31 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
35 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
24 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
34 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
42 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
35 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
23 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
20 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
68 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
16 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
3answers
77 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
23 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 ...