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
15 views

Ladybug walking on a hexagon, mosquito walking on a number line probability question

1) A ladybug is walking at random on a hexagon. The ladybug begins at Vertex A. Each minute, the ladybug moves to ONE OF the randomly chosen TWO vertices adjacent to the one she's currently on. ...
1
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3answers
37 views

How many ways to make a connected graph using 4, 5, 6 edges?

How can/how many ways can you make a connected graph that has 5 vertices using 4, 5, 6 edges? I'm not sure how it would look like for 4 edges. Can you draw a diagram?
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2answers
32 views

Interchanging the order of a double infinite sum

I'm stuck at a proof of Wald's first equation about interchanging the order of a double infinite sum: Suppose $X_n \ge 0$ and $1_{\{\cdot \}}$ be indicator function. $$ \sum_{n=1}^\infty ...
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0answers
14 views

central limit theorem, solving for probability

I am playing a game and am trying to calculate the probability that I will win at 40,000 or more points total, if I play the game 1,000,000 times. The expected value for one game is zero and the ...
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1answer
15 views

Probability: Expectation: indicator RV, what is 1-((N-1)/N))^n?

Say there are N coupon types, you collect n coupons, and what's the expected number of types of coupons? My question is specifically about $1-(\frac{N-1}{N})^n$, the probability of getting a coupon ...
-4
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1answer
46 views

$3$ dimensional light up cube ornament, difficult probability problem

Imagine a $3$ dimensional cube (much like a $4\times4\,(\times4)$ Rubik's cube) except instead of faces with colors, it is clear but with a single light in each of the $64$ little cubes contained ...
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2answers
31 views

What is my probability space and measurable space?

I have the following difference equation $$ \tilde{u}_k = \begin{cases} u_k & \text{if $\gamma_k = 1$, no signal lost} \\ \tilde{u}_{k-1} & \text{if $\gamma_k = 0$, signal lost} \end{cases} ...
-1
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1answer
33 views

Probability of the sum of exponentials being greater than a number

Say we have Xi ~ Exponential(1/3) We add Xi (all independent variables) until we reach a value of 5 or greater than 5 What is the probability of the sum of Xi being greater than 7? I have no idea how ...
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0answers
11 views

Cinlar Ex. 1.15: Trace space of a measurable space.

In constructing the trace space on a subset of a measurable space, it seems one has to assume that the subset is an element of the original measure space's sigma algebra, i.e., measurable in the ...
-1
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0answers
18 views

AI Bayes Network Question? [duplicate]

A) Given this Bayes Net Answer and explain: 1) True or False 2) True or False B) Given this Bayes Net: Answer and explain: 3) True or False 4) True or False
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1answer
27 views

What is the probability of choosing r objects from c different groups when there are m groups of n objects?

Suppose I have m groups of n objects each for a total of nm objects. I am going to choose r of these nm objects. I want to know what the probability is that my r objects come from c different ...
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3answers
39 views

The probability of Breakeven On a Coin Toss Game

I was walking the other day around my work office in NYC and thought of this interesting scenario in a game of coin flips. You have $500 in your pocket. This is your entire life savings. You play a ...
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1answer
17 views

Four dice, probability that difference of some outcomes is equal to others

I roll four dice which gives me outcomes $x_1, ..., x_4$. I want to determine the probability $$P\left((x_2-x_1) = (x_4-x_3)\right)$$ I have already calculated other probabilities in this setting and ...
-6
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0answers
26 views

AI Bayes network? [on hold]

A) Given this Bayes Net Answer and explain: 1) True or False 2) True or False B) Given this Bayes Net: Answer and explain: 3) True or False 4) True or False
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votes
1answer
21 views

Determine the density of sum of three normal variables.

Setting $\pmb{X} = (X_1,X_2,X_3)$ is a properly center normal with covariance matrix $$\begin{pmatrix} a & b & 0\\ b & d & 0\\ 0 & 0 & e \end{pmatrix}$$ Determine the ...
1
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1answer
25 views

distinguishing probability measure, function, distribution

I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function) probability ...
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1answer
25 views

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
-1
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1answer
30 views

probability density function for random variables [on hold]

Suppose $x$ is a random variable with PDF $F(x)$ to be a continuous distribution. What is the probability of obtaining $x=a$ ?
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0answers
16 views

Standard Deviation, Random Sample, Probability Q #2 [on hold]

A. 90% of people eat their eggs with salt, 75% eat their eggs with pepper, and 65% of people put both pepper and salt on their eggs. What is the probability that a person eats his eggs with salt given ...
3
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1answer
20 views

Probability measures and stochastically dependent events

If $P(B\mid A) > P(B)$ and $P(C\mid B) > P(C)$ can I infer that $P(C\mid A) > P(C)$? My suspicion is yes but I don't see how to prove it yet.
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votes
1answer
44 views

Standard Deviation, Random Sample, Probability

Suppose the average person spends \$16 per week on soft drinks, with a standard deviation of \$2.50. If a random sample of 47 people is taken, what is the probability that the mean amount spent on ...
0
votes
1answer
20 views

Constructing a joint distribution given $P(X\in A \mid Y)_\omega$

For random variables $X,Y,Z$, I am given for any measurable set $A$ $$P(X\in A \mid Y)=P(Z\in A\mid Y) \text{ a.s. }\iff (X,Y)\overset{d}{=} (Z,Y).$$ The direction $\Leftarrow$ doesn't seem too ...
2
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1answer
30 views

Consistency vs Inconsistency in a set of sentences: which is more common

I'm curious whether there is any research in the "probability" that a set of sentences in a first-order logic is consistent. Obviously, there are an infinite number of inconsistent sets and an ...
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3answers
22 views

probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
2
votes
1answer
41 views

Is this an upper bound or lower bound?

I came across a probability distribution function in my work, it is however difficult to find in closed form, therefore I am looking to either upper bound or lower bound it. Assuming $a,b,T$ are ...
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0answers
10 views

determine how much probability increase with an added condition

Suppose there are $N$ people and $N$ prizes, and only $M$ out of $N$ are valuable. Every time one person is picked randomly, then he pick one prize randomly as well (this prize/person is then removed ...
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0answers
19 views

Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
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0answers
7 views

Distribution of the ratio of two dependent chi-square

I look for my work the distribution of the ratio of two dependent chi-square variables $X, Y$ with different degrees of freedom for each one. Meanwhile I only found the distibution for the case where ...
2
votes
1answer
29 views

Uniform Distributions in Probability

X, Y, and Z are independent and uniformly distributed over [0,1]. I'm trying to find the distribution of XY by using the joint transformation T = X, W = XY. We haven't learned transformations yet, ...
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0answers
18 views

Determine if the following family of hash functions is universal

Let $H = \{h_1,h_2,h_3\}$ be the family of hash functions defined below, each mapping $\{a,b,c,d,e\}$ to $\{0,1,2\}$. Is $H$ universal? A family of hash functions is universal if $\forall ...
0
votes
1answer
9 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
0
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0answers
13 views

Approximate normal distribution(this is different from what I asked earlier $\log(n)$ is replaced by $\sqrt{\log{n}}$)

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
0
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1answer
21 views

Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
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0answers
7 views

Convergence of third moment in central limit theorem

Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian. If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = ...
2
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2answers
42 views

The limit of an expected value vs expected value of a limit in this betting game

Setting The outcome $X$ of a slot machine takes values 1,2,or 3 with probability $p(1) = \frac{1}{2}$, $p(2) = \frac{1}{4}$, $p(3) = \frac{1}{4}$. We are given 3 for one odds, that is if we bet 1 ...
0
votes
1answer
29 views

Central limit theorem in the setting of Poisson variables

Setting Given $S_{\lambda} \overset{d}{\sim} \operatorname{Poisson}(\lambda)$. Let $G_{\lambda}(t)$ be the distribution function of $\frac{S_{\lambda}}{\lambda}$. I need to determine ...
1
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1answer
22 views

What can you conclude about the first moment of a variable given the 3rd moment exists and is finite

Suppose you are given a random variable $X$ and told that $E[X^3]$ exists and finite. Can you conclude that $E[X]$ exists and is finite? What about $E[X^2]$? How would you argue rigorously whether ...
1
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1answer
13 views

Using chi-square test for statistics with multiple options in one variable

i think that this is more mathematical question, i am doing some statistics on survey. There is questions with multiple choices, so for example, if there are 4 choices, i can pick 1st ,2nd and 4th. ...
1
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1answer
23 views

Random Sample vs Simple Random Sample

I am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...
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1answer
23 views

Probability distribution for putting balls in boxes in a correlated way

I'm looking for help finding a probability distribution: Right now I have a problem where I have N indistinguishable balls, which I need to put into K indistinguishable boxes, each of which can hold ...
1
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1answer
40 views

Using the geometric distribution to find the probability that between 4 and 6 devices will be tested

Quality control tests spark plugs until they find one that doesn't work. If the probability of a spark plug working is 0.99, what is the probability that they will test between 4 and 6 (inclusive) ...
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2answers
23 views

Are these transient or recurrent states in a Markov chain?

I have the following transition matrix for a Markov chain with states $A, B, C, D, E$ $ \left| \begin{array}{ccc} 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & ...
8
votes
1answer
58 views

probability that no two spiders end up at the same vertex?

Eight spiders are located on the eight vertices of a cube. When a bell rings, each spider moves (at random, independent of the others) to an adjacent vertex. What is the probability that no two ...
1
vote
1answer
21 views

We have an urn with 5 blue balls and 15 red balls.

We remove 7 without replacement. Let R be the number of red balls removed and B the number of blue balls removed. Do you expect R and B to be positively correlated, negatively correlated, or ...
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0answers
31 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
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1answer
31 views

Distribution problem where |a|, |b|, |c|, and |d| are at most 10. Check my work?

How many ways can a+b+c+d=18, where a,b,c,d are integers such that $|a|,\ |b|,\ |c|,\ |d|$ are each at most 10? This is what I have so far. If all four numbers have the restriction -10 =< a, b, ...
1
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1answer
32 views

Selecting n matches from two pockets.

Setting An eminent mathematician fuels a smoking habit by keeping matches in both trouser pockets. When impelled by need he reaches a hand into a randomly selected pocket and grubs about for a match. ...
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2answers
35 views

Total possible game scenarios

I am trying to figure out every possible scenario of every team in a league either winning losing or tying given the amount teams and weeks left in the season. For instance, the possible scenarios ...
1
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0answers
20 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
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1answer
22 views

Conditional distribution of geometric variables

Setting Suppose X1 and X2 are independent with the common geometric distribution w(k; p). Determine the conditional distribution of X1 given that X1 + X2 = n. Solution My argument is $$\Pr[X_1| ...