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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-4
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0answers
53 views

Affine function of cdf [closed]

Suppose a random variable X has cdf FX(·). Express the cdf of the following random variables: X + b aX + b |X| max(X,0) Could someone show me how to use the given random variable X and its ...
0
votes
0answers
48 views

What is the probability that ant will be on square $S$

Plane is filled with squares which sides are parallel and have equal length. Each square is black or gray. An ant is placed on one black square. Ant can walk only on black squares and cannot walk ...
0
votes
1answer
22 views

Independence of two random variables $\xi_1,\xi_2$

Two-dimensional random variable $ p_{\xi_1\xi_2}(x_1,x_2)=\begin{cases} \frac 1 {6\pi},& \frac {{x_1}^2} 9+\frac {{x_2}^2} 4 \leqslant 1,\\ 0,&\frac {{x_1}^2} 9+\frac {{x_2}^2} 4 > 1. ...
0
votes
1answer
23 views

Find approximation for size of population over time

Assume you start with a population of an objet of size $1$. Assume that a new objet of size $1$ is born at each date and that existing objects double in size in each period. Over time the sequence of ...
0
votes
0answers
17 views

Conditioning on a conditioned event

I am calculating the probability of error in using a particular transmission scheme and would like to know if the below is correct: $$ P(A|B)=P(A|(C|B))P(C|B)+P(A|(C'|B))P(C'|B) $$ What does it ...
2
votes
1answer
27 views

Probability of getting exactly $\alpha$ “A” runs and $\beta$ “B” runs in a randomly drawn word

Let $a, b \in \mathbb{N}$. Each "word" that we can form using exactly $a$ times the letter $"A"$ and $b$ times the letter $"B"$ is written onto a card. Next, one of these $\pmatrix{a + b \\ a}$ cards ...
1
vote
3answers
30 views

Simple Conditional Probability

Among 60 automobile repair parts loaded on a truck in San Fransico , 45 are destined for seattle and 15 for vancouver. If two of the parts are unloaded in portland by mistake and the selection is ...
0
votes
1answer
62 views

Help with this question from my textbook

Hello I've been battling with this particular question from my statistics textbook for hours. Can someone kindly help with this. Note: it is not an assignment question. I'm solving all questions in ...
0
votes
0answers
17 views

Is there a theorem or law stating that the expected value of a symmetrical distribution across $f(X)$ will equal $f(\bar{x})$ iff $f'=c$ across X?

I have noticed that the expected value of a symmetrical distribution across $f(X)$ will equal $f(\bar{X})$ only if $f'$ is constant across X. For example, consider a uniform distribution X ranging ...
1
vote
1answer
13 views

A particular implication of convergence in probability

Suppose that $(X_n)_{n\in\mathbb N}$ and $X$ are random variables on a probability space $(X,\Sigma,\mathbb P)$ and $X_n\overset{\mathbb P}{\to}X$. That is, for each $\varepsilon>0$, ...
2
votes
2answers
46 views

Combination and Probability

There are n students and n+2 different gifts. Each student have to receive 1 gift package. How many ways can we give out all the gifts. ...
2
votes
1answer
34 views

A question about the conditional PDF?

Here is the question: In this problem: (i) X is a (continuous) uniform random variable on [0, 4]. (ii) Y is an exponential random variable, independent from X, with parameter λ = 2. Find the PDF ...
1
vote
2answers
33 views

combining probibilities

I am working my way through Probability and Statistics for Computer Scientists 2nd ed. And am stuck on how to answer the following question: 2.10 Three computer viruses arrived as an e-mail ...
0
votes
0answers
29 views

Find probability distribution

Cumulative distribution function $F_\xi(x)$ continuous at $0$. Find the probability distribution random variable $ \eta = \begin{cases} \frac \xi {|\xi|},&\xi \neq 0,\\ 1,&\xi = 0. ...
0
votes
2answers
28 views

Sum of uniform random variable and non-uniform random variable [closed]

Let $G=\mathbf{Z}/p \mathbf{Z}$ where $p$ is prime, $X\in G$ be a uniform random variable and $Y\in G^{*}$ be any random variable. Is it possible to have $Z=X+Y \in G$ with a uniform distribution? ...
0
votes
0answers
28 views

Probability//Uniform Distribution [closed]

It is believed that the time $X$ (min) for a lab assistant to prepare the computer for a certain experiment has a uniform distribution with $A = 25$ and $B = 35$. How can I determine the density of ...
1
vote
4answers
31 views

A team squad combination and probability problem

A team of 11 is chosen randomly from a squad of 18. Two of the squad are goal keepers and one of them must be chosen. If neither of the goalkeepers is captain or vice captain, what now is the ...
1
vote
0answers
19 views

Markov property of ito diffusion [duplicate]

Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$. In this ...
-1
votes
0answers
14 views

applying constraints on parameter in MATHEMATICA [closed]

How does one apply constaints on parameters in MATHEMATICA? F(x)=[1+a*(1+x^b)^c]^a. here x lies between 0 to inf and parameters a, b, c are greater than 0. how does one simulate this cumulative ...
0
votes
3answers
30 views

Probability One of the three carries the number 5, and the other two numbers Smaller?

10 people playing social games. Each participant has a number 1-10. 3 players randomly selected. What is the probability One of the three carries the number 5, and the other two numbers Smaller? ...
0
votes
1answer
21 views

Probability for random vector given probability distribution [closed]

Given the following probability distribution: $f(x,y) = \begin{cases} xe^{-x-y}, & x,y>0 \\[2ex] 0, & \text{elsewhere} \end{cases}$ compute $P(X \le Y)$. I know that the result is $1/4$, ...
1
vote
0answers
21 views

Filtration of path space

Let $W\left(M\right)$ be the path space of $M$. An element of $W\left(M\right)$ is a continuous map $x:\left[0,\infty\right)\to M$ (with some further technical details). I've been trying to determine ...
0
votes
1answer
28 views

Bernoulli scheme

Cable link has $k$ channels that connect two cities where $n$ subscribers. Each subscriber uses telephone in average $l$ min/hour. Find probability that there will be trouble-free service ...
-2
votes
1answer
30 views

Calculating the probability of passing all courses [closed]

We have five course: $A-E$. Based on past performance, we know that the rough probabilities of passing a course are: $A: 0.85, B: 0.95, C: 0.48, D: 0.94, E: 0.38$ What is the probability that a ...
0
votes
2answers
32 views

A bag contains 6 white buttons, 2 blue buttons. What is the probability that exactly one of the 3 buttons taken out is blue?

A bag of spare buttons contains 6 white buttons and 2 blue buttons. 3 buttons were taken out of the bag at random without replacement. What is the probability that exactly one of the 3 buttons is ...
-2
votes
0answers
32 views

Derivation of pdf from the function of random variables [closed]

Let $A_{i}$ and $B_{i}$ ($i=1,...,K$) be the random variables of which pdf/cdf are known to us. And, there is a function of random variables, $C=\max({A_{1}+B_{1}, A_{2}+B_{2}, ..., A_{K}+B_{K}})$. ...
1
vote
1answer
18 views

Radio communications system - Uniform density question [closed]

Full question: In a radio communications system, the phase difference $X $ between the transmitter and receiver is modeled as having a uniform density in $[—\pi, +\pi]$. Find $P(X \le 0)$ and $P(X \le ...
0
votes
1answer
31 views

A machine produces memory sticks of varying lengths…

Full question: A machine produces memory sticks of varying lengths, distributed uniformly between 2 and 12 mm. Memory sticks longer than 10 mm do not meet the design criterion and must be scrapped. ...
1
vote
2answers
31 views

Finding large deviation bound for binomial distribution

$S \sim Binomial(n, p)$. $\forall a > p$, find large deviation bound for $P( S \geq an)$ In the book, the large deviation bound definition is as follows: $\phi(t)$ is finite for some $t > 0$, ...
0
votes
1answer
16 views

Probability of fulfilling an order using specific number of packages

Generic problem: Given an order for X unique units, what is the probability of being able to fill that entire order using Y packages (or locations), if the probability of finding each unique unit in ...
0
votes
1answer
26 views

Why is the Expected Value not good for this type of question relating to frequency?

Number of Persons Living in Household | Relative Frequency 1 | 0.36 2 | 0.28 3 | 0.14 4 | 0.15 5 | 0.05 5(use 6.3 for avg) | 0.02 Predict the expected population ...
1
vote
1answer
25 views

Finding the Probability of B, knowing the Probability of B | A and B|A'

For a Problem with Events A and B, I know: $P(A)$, $P(B|A)$, and $P(B|A')$ This has come up in a a couple different problems in my homework, and the provided solution always involves the following ...
0
votes
1answer
14 views

How do you compute multivariate normal distribution probabilities?

Given a vector $$ X = (X1,X2,X3)^t $$ which is multivariate normal with mean 0 and covariance matrix $$ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 3 ...
0
votes
0answers
34 views

Expected random word length found in random text

Introduction There is a randomly generated text of (M) characters. There are (q) different characters. If I generate another random text with the same (q) and of length N value over and over again ...
0
votes
3answers
39 views

Expectation Value of a Multiset

Imagine that I have $k$ balls randomly distributed (uniformly) among $n$ boxes. I.e., with repetition. How could I calculate the expected number of balls in a randomly chosen box?
1
vote
1answer
18 views

What is your expected waiting time if limousine inter-departure times follow an exponential distribution?

Limousines depart from the railway station to the airport from the early morning till late at night. The limousines leave from the railway station with independent inter-departure times that are ...
1
vote
1answer
70 views

Random Variables in a Uniform Probability Space

Suppose that $\Omega = \{1,2,3,4,5,6\}$ is a uniform probability space. Now, let $X(\omega)$ and $Y(\omega)$, for $\omega \in \Omega$, be random variables defined as: $$\begin{array}{|c|c:6c|} ...
2
votes
1answer
24 views

Expectation of the fraction a random function covers its range

Preamble: The number of onto functions from a set of $m$ elements to a set of $n$ elements is, as stated in this answer, computed as follows: $$n!{m\brace n}\;.$$ Now, let's count the number of ...
0
votes
2answers
42 views

Show $(X_n+a)^2$ is a submartingale

Let $(X_n)$ be a martingale, and let $EX_n^2 < \infty$ - then I am told to show $E(X_n+a)^2 $ is a sub martingale. I wrote $$(X_n+a)^2 = ((X_{n-1} + a) + (X_n - X_{n-1}))^2 $$ then $$E((X_n+a)^2 | ...
1
vote
2answers
83 views

Expected number of women sitting next to at least one man?

There are $10$ seats, $5$ men and $5$ women who randomly occupy these seats. I have to calculate the expected number of women sitting next to at least one man. My attempt: I defined a random variable ...
0
votes
0answers
20 views

Shatter coefficient and VC dimension of a grid in $R^d$

Given $\epsilon>0$, partition the cube $[0, 1]^d$ with square of side length $\epsilon$. The total number of square in the partition is $$ N = \left(\frac{1}{\epsilon}\right)^d. $$ What is the ...
1
vote
1answer
24 views

compute probability density function of a bivariate function without sampling

Suppose $X_1 \sim f_{X_1}(x_1)$, $X_2 \sim f_{X_2}(x_2)$ are random variables with known probability density function. Is there any way to compute the probability density function of a bivariate ...
0
votes
0answers
26 views

How to prove this continuous martingale converges?

Suppose $B = (B_t, t \geq 0)$ is standard Brownian motion. Let $M^\lambda_t := \exp(\lambda B_t - \frac{\lambda^2 t}{2})$ (I have previously shown that this is a martingale). How do I prove that $$ ...
4
votes
2answers
38 views

posterior probability of bag given ball (evidence)

Question: Given the distribution of the coloured balls in three different bags: - Bag A: 1 Red 2 Black 2 Blue - Bag B: 2 Red 4 Black 4 Blue - Bag C: 10 Red 2 Black 3 Green we carry out ...
3
votes
1answer
15 views

co-variance between a sample from normal distribution and the sample mean?

$X_1$ is a sample from a normal distribution with mean$=\mu$ and variance $= 1$. The joint distribution of $X_1$ and the sample mean is bivariate normal. I need to find the conditional distribution of ...
1
vote
1answer
55 views

X and Y have joint density function c/x^3

Suppose X and Y have joint density function: \begin{equation} f (x,y) = \begin{cases} c/x^3& \text{} x < y < 1\\ 0 &\text{otherwise}, \end{cases} \end{equation} where c is a ...
-1
votes
1answer
18 views

Conditonal Density Function

$$f_{X,Y}(x,y)= \begin{cases} (x-y) & 0 \lt x \lt 1, & -1 \lt y \lt 0 \\ 0 & \text{elsewhere} \end{cases}$$ How can I find the conditional density function $Y$ $$f_Y(y | ...
2
votes
1answer
34 views

Borel-Cantelli exercise

I'm stucked with this exercise. Let $X_1,X_2,\ldots$ be i.i.d. random variables with $E(X_1)=0$ and $Var(X_1)=\infty$ Prove that$$P(\limsup\limits_{n\to\infty}\{|X_n|\geq \sqrt{n}\})=1$$ I need to ...
1
vote
0answers
20 views

Generating a predictive distribution from a small sample

I have an oracle that gives numbers from an approximately-normal distribution, but I do not know the mean or variance of the distribution itself. (I do happen to that the oracle does not produce ...
1
vote
1answer
32 views

Convergence of series of random variable without distribution

I'm trying to solve the following task and I'm struggling very much. I don't know if it is correct what I did so far. Let $(X_n,n\geq1)$ be a sequence of independent random variables such that ...