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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3
votes
1answer
10 views

Choosing exactly 2 damaged pieces (Probability)

From $27$ pieces of luggage, an airline handler damages a random sample of $4$. The probability that exactly one of the damaged pieces of luggage is insured is twice the probability that none of ...
0
votes
3answers
28 views

Probability of winning a rigged coin-flipping game

Betsy and Katie are playing a game with an unfair coin. The coin is rigged to come up heads with probability $\frac35$ and tails with probability $\frac25$. Betsy goes first. The two take turns. The ...
5
votes
4answers
60 views

How can I write this power series as a power series representation?

How can I write this power series ($1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8....$) as a power series representation (like $\dfrac{1}{1-x}$ or something neat like that)?
0
votes
2answers
32 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 ...
1
vote
1answer
25 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
2
votes
5answers
65 views

Jamie rolls a die multiple times. find the probability that she rolls her first 5 before she rolls her second even number

Jamie rolls her fair 6-sided die multiple times. Find the probability that she rolls her first 5 before she rolls her second (not necessarily distinct) even number? This is what I have so far... ...
0
votes
1answer
19 views

Using Poisson distribution to find the probability that the interval between arrivals exceeds some value

Suppose we have a helpdesk with tickets arriving at a rate of three per min. Tickets arrival follow a Poisson distribution. How someone can calculate: a. The probability of the time between the first ...
-7
votes
0answers
28 views
1
vote
0answers
10 views

the quantile to quantile plot for the discrete law

the quantile to quantile plot works good for the continuous law. But I don't know why it doesn't work for the discrete law, example the law Poisson. Please help me
11
votes
5answers
1k views

How to calculate the probability of rolling 6 at least 5 times in a row, out of 50 tries?

If I roll the dice 50 times, how do I calculate the chance that I will roll 6 at least 5 times in a row? Why this problem is hard With 5 tries this would be easy: take $(1/6)$ to fifth power. With ...
1
vote
3answers
53 views

How to find the number of possible outcomes of 10 games between 20 teams?

Hi I am looking for an equation to find possible combinations in a non repeating format with a twist. Here is the example: There are 10 games between 20 teams. I have to chose 5 winners but ...
0
votes
4answers
49 views

Expectation of non-negative random variable

Let $X$ be a non-negative random variable. In a proof for $E[X]=\int_0^\infty P(X>t)dt$ from the answer of this question, we use Fubini for the middle quality. Why do we need $X$ to be ...
1
vote
2answers
31 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 ...
0
votes
2answers
36 views

Ladybug walking on a hexagon probability question [on hold]

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. Find ...
0
votes
1answer
31 views

The number of self-avoiding paths in the plane of length $k$

The number of self-avoiding paths in the plane of length $k$ is at most $4 \cdot 3^{k-1}$ according to this. Why? My immediate thought: four options for the first move and always three choices after ...
0
votes
1answer
20 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
-1
votes
1answer
61 views

$3$ dimensional light up cube ornament, # of rows/cols/diags in/on a cube

Imagine a $3$ dimensional cube (much like a $4\times4\,(\times4)$ Rubik's cube) except the planes of the cube cannot be twisted individually and instead of faces with different colors, it is clear ...
0
votes
1answer
18 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 ...
2
votes
0answers
65 views

If $x \sim U(Z_n^*)$ then $x^2 \pmod n\sim U(QR_n)$?

Define: $Z_n^*=\{x \in Z_n | \operatorname{gcd}(x,n)=1\}$ $QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$ How can I show that $x \sim U(Z_n^*) \implies x^2 \pmod n \sim U(QR_n)$? Thank ...
0
votes
2answers
41 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 ...
2
votes
1answer
20 views

Finding the PDF from the CDF where the CDF is not differentiable at some point

I got the following problem: Let $X$ be a continuous random variable with $CDF$ denoted $F_X$ defined as follows: $F_X(x)= \begin{cases} 1-x^{-4/3}, & x\in[1,\infty) \\ 0, & x\in ...
0
votes
1answer
23 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 ...
-5
votes
0answers
47 views

Where can I find consultants on Long Lead Coin Tossing Experiments? [on hold]

Where can I find consultants on Long Lead Coin Tossing Experiments? I am specifically interested in the phenomenon of Long Leads, which was discussed in Feller's classic text. I want somebody to ...
1
vote
1answer
49 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 ...
0
votes
0answers
15 views

Find probability density function of random vector

Random vector has continuous distribution of setA={(u,v), v>=0, u+v<=1, v-u<=1}. I need to find joint probability density function of this vector. In my ...
0
votes
0answers
10 views

Limit distribution on return time $\tau = \inf\{k: X_k = X_m \text{ where }m<k\}$ [on hold]

Suppose there is a stochastic process ${X_i}_{i=1}^n$ where $X_i$ is distributed normally over $\{1,\dots,n\}$. As $n\rightarrow \infty$, the probability that any one value is repeated should go to ...
0
votes
0answers
4 views

What is the formula for the 2-sample Anderson–Darling upper tail test?

There are computationally simple formulas for the Anderson–Darling test between an analytic distribution and an empirical distribution, as well as for the Anderson–Darling upper tail test (again ...
5
votes
1answer
339 views

Memoryless property and geometric distribution

Suppose $X$ is a random variable taking values in $\mathbb N_0$ with the memoryless property,i.e., for each pair of number $s,t \in \mathbb N$, $$P(X\geq s+t\mid X>t)=P(X\geq s)$$ Show that a ...
0
votes
2answers
398 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
0
votes
2answers
49 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} ...
0
votes
3answers
50 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?
1
vote
2answers
41 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 ...
2
votes
1answer
31 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, ...
1
vote
1answer
17 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 ...
0
votes
1answer
39 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 ...
2
votes
2answers
99 views

Expectation of Random variable Exam Question

Suppose for a random variable $X$ it is given $P(X \ge a)=1-\frac{1}{4}a^2$ , for $0\le a\le 2$. what is the expectation of X? Correct answer: $\frac{4}{3}$ I have difficulty solving the problem ...
0
votes
1answer
24 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
vote
3answers
44 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 ...
2
votes
2answers
59 views

Probability for having consecutive success in an experiment

A friend asked me the following question: "In an experiment, we are tossing a fair coin 200 times. We say that a coin flip was a success if it's heads. What is the chance for having at least 6 ...
0
votes
1answer
24 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 ...
3
votes
1answer
2k views

Probability of finding at least k consecutive heads in N coin tosses?

There are quite a few topics on this question already but I couldn't find a well-explained solution. Please point me towards some relevant literature or theory to analyze this problem. $K$ ...
0
votes
0answers
21 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
0
votes
0answers
14 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 ...
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 ...
-5
votes
0answers
28 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
1
vote
1answer
26 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 ...
0
votes
1answer
32 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.
3
votes
1answer
295 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
-1
votes
1answer
46 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 ...