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 (2)

0
votes
1answer
34 views

Help me find $P(A \cap B')$ under the given conditions

I was assigned the task to solve this problem by mathematics teacher which I can't solve because it doesn't make sense to me (I think that it is impossible to solve it). There was an error please try ...
0
votes
0answers
20 views

Is it necessary to normalize likelihood within an event space before further multiplication among events?

Say I have observed data, and parameters $A,B$: Parameter $A$ contains possible values: $a_1,a_2,a_3$ Parameter $B$ contains possible values: $b_1,b_2,b_3$ Now, assume I know the likelihood of ...
0
votes
1answer
12 views

$\Pr(X+Y\geq1)$

Two random variables X and Y have the following joint pdf: $$f_{X,Y}(x,y)\begin{cases}10x^{2}y & 0<x<1,0<y<x\\0 & \text{otherwise}\end{cases}$$ I am asked to find the marginal pdf ...
2
votes
2answers
28 views

What are the odds that two people are friends in a network of 20 people?

If person $A$ has 10 friends and person $B$ has 5 friends, and they are in a network of 20 people, what are the odds that persons $A$ and $B$ are friends? I first thought to divide into cases ...
2
votes
2answers
31 views

Probability mean,variance and standard deviation formula confusion.

I have a confusion in the formula attached. Why and how are the two formulas equivalent ? sigma in the image is the standard deviation of a distribution...
1
vote
2answers
301 views

Probability Dealing with Cards

Here is the story. You are dealt 8 cards from a well shuffled deck of 52 cards. What is the probability of getting 4 queens and 4 kings? I understand the probability is quite small but how do we set ...
4
votes
0answers
165 views
+100

A fun card game involving probability, getting all 13 ranks (any suit(s)) vs. 5 in a row of red or black.

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, shuffled well before each hand is drawn, and randomly draw cards from it one a time ...
0
votes
2answers
1k views

What is the expected number of times a 6 appears when a fair die is rolled 10 times?

Ok, so I think I have a working solution to this problem. Heres how I would solve it: so you look at a 6 appearing as a success and everything else as a failure. So from here you can you use the ...
0
votes
1answer
27 views

Significance level for a hypothesis test for a linear regression

Consider linear regression model $Y_i=a+b\cdot x_i+\epsilon_i$, $i=1,2,3,4,5$, where $a,b\in\mathbb{R}$ are unknown and $x_1=x_2=1,x_3=3,x_4=x_5=5$, $\epsilon_i$ are iid, normally distributed with ...
2
votes
2answers
47 views

Find the probability of solutions of an equation.

Let $x+y+z=20$. What is the probability that all the solutions are distinct? (No two variables have the same value). Assuming that the solutions are only positive integers or zero. I have tried- ...
1
vote
1answer
52 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
1
vote
2answers
33 views

Showing that the Lindeberg CLT Condition Holds

Suppose we have a sequence of random variables, $\{X_{n}\}_{n\geq 1}$ satisfying: $\mathbb{P}(X_{j} = 2^{j}) = \mathbb{P}(X_{j} = -2^{j}) = \frac{1}{2}$ Then is it true that the CLT holds? Or ...
4
votes
1answer
39 views

Roll eleven dice such that the product is prime

So the problem is: What is the probability of rolling eleven dice such that their product is prime. The dice is numbered from 1 to 6 and there is an equal chance of getting each number. So in order ...
-2
votes
0answers
13 views

In tennis, the probability of a player winning a point on serve given serve statistics. [on hold]

How can I calculate the probability, $p$, of a player winning a point when serving given: The percentage of first serves that the player gets in. (I'm not sure this is relevant/needed). The ...
0
votes
2answers
14 views

Probability Multivariate Distributions

A computer generates two independent fixed numbers from a uniform distribution on the range [0,100].Calculate the probability that the first fixed number exceeds the second by at least 20. I'm ...
3
votes
1answer
26 views

Probability and cards

A box contains 900 cards enumerated from 100 to 999 (Each number appears once and just in one card). I took some random cards without looking at them and calculated the additions of the digits in each ...
0
votes
1answer
448 views

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f

For each of the following, determine the constant c so that f(x) satisfies the conditions for being a p.m.f. for a random variable X. c) f(x) = x/c, x = 1,2,...,n d) f(x) = c/(x+1)(x+2), x = ...
-4
votes
1answer
10 views

Maximizing Varience of Independent Random Variables [on hold]

Suppose X and Y are independent mean 0 random variables, with positive variances a and b, respectively. Find the value of c that minimizes the variance of cX+(1-c)Y?
2
votes
1answer
37 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
0
votes
1answer
13 views

Finding the y-coordinate of the peak in a gaussian distribution?

First off all, my general understanding of gaussians is not very good, and I'm having issues getting my head around this because I cannot find an explanation of them I can understand. I'm working ...
0
votes
0answers
13 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
0
votes
0answers
10 views

Random variables set representation in the sample space

Consider that I have two Random variables $ X : \Omega \rightarrow \mathbb{R} \space , Y : \Omega \rightarrow \mathbb{R}^d$ belonging to the same sample space and a measurable function $\space f : ...
-1
votes
0answers
21 views

Inequality with poisson distribution [on hold]

Let $r>1$ and $X \sim Poiss(\lambda)$. Prove that $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ Does this inequality hold for $r>0$ ?
2
votes
1answer
483 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
3
votes
1answer
50 views

Probability - Poisson arrival of rain

I'm trying to solve this Poisson problem. A rain shower lasts 10 minutes and in that time deposits $10^6$ raindrops over 100 $m^2$. a) What is the probability of at least one drop landing in 1 $cm^2$ ...
1
vote
1answer
24 views

About the equivalence of two asymptotic probabilistic statements

Let $g(n)$ be some monotone increasing function of naturals, and let $X_n$ be a sequence of positive random variables. Consider the following two claims: Claim 1. $\exists f=o(g(n)),\ ...
0
votes
2answers
29 views

How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes so that each of the boxes contain at least 1 object?

How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes so that each of the boxes contain at least 1 object? Can anyone tell me how should I approach this ...
1
vote
0answers
23 views

Random variable: $X\sim Normal(m, {\sigma}^2)$, find the characteristic function of $X^2$

Is it possible, knowing that $X$ is a random variable with normal distribution( with parameters $(m, {\sigma}^2)$), to find the characteristic function of $X^2$ ? What I thought is: Since: $\phi(X) ...
-1
votes
1answer
13 views

Find marginal probability distribution of $ X$?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X= 5\ mL,\ \sigma_Y= 2\ mL, \ \mu_X= 120\ mL, \ \mu_Y= 100\ mL$ and $\rho = 0.6.$ How do I find the marginal probability distribution of ...
0
votes
1answer
26 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
2
votes
2answers
326 views

Two Probability Questions

Studying for a final: 1) Suppose we have four indistinguishable red balls, 6 indistinguishable blue balls, and 2 indistinguishable green balls. How many different color patterns can be obtained by ...
1
vote
1answer
21 views

Determine the probability density function of…

Let $X$ be a random variable with normal distribution with parameters: $$m = 1$$ and $$\sigma = 2$$ How can the probability density function of $$Z = -\frac{\ln |X|}{3}$$ be determined?
2
votes
3answers
65 views

Can the probability of an event be an irrational number?

I am wondering whether it is possible to construct an experiment, where the probability of occurrence of an event comes out to be an irrational number.
-3
votes
0answers
28 views

in how many ways i can put three things in two bags [on hold]

I have two oranges,one apple and one banana. i want to put two of them at a time in two bags having one(one froot at max in one bag) each. There are two oranges and they are indistinguisable(they are ...
0
votes
0answers
15 views

Serial Number in a Geometric Distribution

I won't bother posting the whole exercise.Basically, we've got 2000 pc's and 12 of them are malfunctioning. At some point, the exercise writes: We choose the pc's until we find a malfunctioning ...
1
vote
1answer
10 views

Conditional Probability of Poisson Variables

I have two independent Poisson variables $X$ and $Y$ with parameters $\lambda$ and $\mu$, respectively. I defined $Z=X+Y$ and found that $Z$ is also Poisson-distributed with parameter $\lambda + \mu$. ...
0
votes
1answer
17 views

Width of Gaussian distribution from N trials of coin tossing

What is the width of the Gaussian distribution that is generated from performing $N$ trials of coin tossing? Example: In a trial of 1000 tosses of a coin, $P(H)=0.5$ and $P'(H)=0.5$, where $H$ refers ...
0
votes
0answers
16 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
0
votes
0answers
18 views

Find distribution of rv X_N where N is independent rv and each X_i~exp(\lambda_i)

First time attempting to use MathJax... Excuse my messy question. Question reads: Let $X_1,X_2,\ldots,X_n$ be independent random variables such that $X_i\sim\exp(\lambda_i)$ such that if $i\neq j$ ...
9
votes
2answers
137 views

At time n, randomly choose a natural number ≤n. How long is it until a single number is chosen three times?

To clarify, the number ≤n is chosen uniformly at random at each step, and n chooses from the natural numbers beginning with 1. I wish to determine the expected value of $n$ at which a natural number ...
0
votes
1answer
35 views

Proof of infinite monkey theorem.

I was just wondering, does the infinte monkey theorem also has a proof? And why is this called a theorem? It is sheer common sense. And what are its applications. I have heard about PHP and IEP and I ...
1
vote
1answer
45 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
0
votes
0answers
12 views

Sample complexity of coin bias problem

I am reading a paper involving learning in Multi-armed bandit case (its okay if you don't know what that is. Just trying to give context here.) To give sample complexity lower bound, they reduce their ...
2
votes
1answer
320 views

Looking for first course textbooks on probability and statistics for math majors

I am taking a probability and statistics course soon and would like to find a text book that is targeted more towards math majors rather than engineers (which is what this class is). The book my ...
0
votes
1answer
15 views

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$, $U = X+Y$. What are $E[X|U], E[Y|U]$?

$X \sim N(0, \sigma_1^2)$, $Y \sim N(0, \sigma_2^2)$. X, Y are independent. $U = X+Y$. What are the values of $E[X|U], E[Y|U]$? I understand $E[X|U] + E[Y|U] = U$, but I'm not sure how to move ...
0
votes
2answers
21 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
-5
votes
0answers
48 views

REALLY tricky Probability question [on hold]

Here is a board game. $$ \longleftarrow \text{left} \qquad\qquad\qquad \text{right} \longrightarrow$$ $$\bigg| \text{win} \bigg| -2 \bigg| -1 \bigg| \text{start} \bigg|\ 1\ \bigg|\ 2\ \bigg| ...
13
votes
1answer
190 views
+100

Show two random variables have same distribution

Let X, Y be two non-negative random variables satisfying the condition $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. How can one show that X and Y are equal in ...
1
vote
2answers
37 views

Adding probability of multiple dice rolls

Can anyone tell me what are the odds that stage 4 will be reached?: Stage 1: roll a 20 sided die results must be 13 or lower Stage 2: roll a 20 sided die results must be 13 or lower Stage 3: roll ...