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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0answers
14 views

How to compute the L^2-distance of a given function to the set of Gaussian functions

I am faced with the following question: given a probability density function $f$ over $\mathbb{R}$ with $\int_{\mathbb{R}}f(x)x^2dx=\sigma^2$ given, I am trying to find the "nearest" Gaussian to ...
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1answer
437 views

A group of thirty people is selected at random. What is the probability that at least

1-A group of thirty people is selected at random. What is the probability that at least two of them will have the same birthday? My answer is $$\frac{_{365}P_{30}}{365^{30}}.$$ When I calculate this ...
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0answers
18 views

Half-life Probability

When trying to answer a probability question a while back, I came upon a fundamental issue embodied by the following problem. Suppose there is a block of 200 atoms, each with a half-life of 60 years. ...
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1answer
27 views

interesting problem of probability

A flea is jumping on the vertices of a square ABCD. It starts at vertex A, and in each jump he moves to an adjacent vertex with a probability of 1/2 for each. The flea will only stop when it has ...
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2answers
26 views

Find the CDF of a function given its PDF

The probability density function of the random variable X is as follows $f_{X}(x) = \begin{cases} 1/4, & \text{if 0 < x < 1} \\ 1/4, & \text{if 2 < x < 4}\\ 1/4, & \text{if 6 ...
0
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1answer
34 views

Test a nuclear plant in two rounds

Before you turn on a nuclear power plant, it is subjected to a first round of three independent tests. Each test has a probability of failure of $p$. After performing these tests, If all three ...
22
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6answers
11k views

Probability that n points on a circle are in one semicircle

Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
1
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1answer
651 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
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3answers
20 views

Will the remainder of multiple dice rolls be fair if at least one roll is performed fairly?

Suppose Alice and Bob are playing a dice game. They each hold a six sided die and a cup. They shake their die in the cup, flip the cup on the table and reveal the roll at the same time. The result is ...
3
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0answers
23 views

What is the limit distribution of $\frac{S_{N_n}}{\sigma \sqrt{a_n}}$ as $n\rightarrow \infty$.

Let $X_1, X_2, X_3,...$ be iid with $\mathbb E[X_i]=0$ and $\operatorname{Var}[X_i]=\sigma^2>0$, and let $S_n = \Sigma_{i=1}^{n} X_i$. Let $N_n$ be a sequence of integer valued random variables ...
0
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0answers
12 views

n is the number of Bernoulli trials with success p. Let $X_i$ be the number of attempts until success. What is the joint probability function?

n is the number of Bernoulli trials with success p. Let $X_i$ be the number of attempts until success. What is the joint probability function where $i=1,2$. Well let's figure them out separately ...
0
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1answer
14 views

2 Cards are picked from a deck without replacement. Let X= number of aces, and Y= number of kings. Find the joint probability function.

2 Cards are picked from a deck without replacement. Let X= number of aces, and Y= number of kings. Find the joint probability function (in a 3x3 table) X and Y are both discrete random variables ...
0
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0answers
14 views

About the expected transitions in Markov Chain

The problem is here: The given answer is here: K = $2+ X_1 + X_2$, where $X_1$ and $X_2$ are independent exponential random variables with parameters $2/3$ and $3/5$. $$ E[K] = 2=2+1/p_1 +1/p_2 = ...
0
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2answers
15 views

Probability: Reading tables and using the data from them?

Alright probability is not as hard as I imagined yet I strugle with reading tables and applying them to the formulas. The question bellow has a table with 3 rows and 3 collumns and I am asked to see ...
1
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1answer
38 views

A silly question regarding a badly written exercise for probability equations.

I am doing some exercises and this silly question is bothering me even though I am familiar with probability theory and Bayes law but this question is written in a rather peculiar manner I have no ...
0
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0answers
8 views

First Moment Method

Assign one of $k$ code numbers uniformly and independently at random to $n$ secret agents. Use the first moment method to show that if $k = ω(n^2)$ then with high probability no two agents receive ...
0
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0answers
49 views

Joint probability for results of a pair (biased) coins [on hold]

If there are two different coins (coin A and coin B) with probabilities of tails equal to pA and pB respectively, and I toss coin A n times and coin B m times (in no specific order), what are the ...
0
votes
1answer
16 views

52% of people want to ban smoking Use the normal approximation to estimate that over half of a sample size $n$ support the the ban

52% of people want to ban smoking. Use the normal approximation to estimate that more than half of a given sample size $n$ support the the ban. q=1-0.52=0.48 For $n=11, 101, 1001$ Are these steps ...
1
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0answers
25 views

Optimal choice for the values of money units

I just thought about how to find the optimal values for money units, given that you want your currency to come in $n$ different values (e.g. Euros come in 7 values for bills and 8 values for coins, so ...
0
votes
1answer
14 views

Finding Transition Probabilities using Metropolis Hastings

I want to find the $4$x$4$ Probability Transition Matrix under the temperature parameter T=2 of Metropolis Hastings. I know that, if x and y are neighbors, $p(x,y) =$ $$ f(x) = \left\{ ...
0
votes
1answer
22 views

Let $ 0 \lt \alpha \lt 1$. $z_a$ is a solution to $\Phi(z_a)=\alpha $.

Let $ 0 \lt \alpha \lt 1 $. $z_a$ is a solution to $\Phi(z_a)=\alpha $. 1.) What is the relation between $z_a$ and $z_{(1-a)}$ 2.) Find $z_a$ (with an error that does not exceed 0.01) for the ...
0
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2answers
21 views

Central limit theorem on packs of variables

I'm trying to solve the following exercise: Let $\mu$ be a probability distribution on $\mathbb{R}$ having second moment $\sigma^2<\infty$ such that if $X$ and $Y$ are independent with law ...
3
votes
3answers
189 views

Expected rolls to get 3 of any number

Suppose I am rolling a die repeatedly, and I keep a tally of how many times each number has come up. As soon as a number has come up 3 times, the game is over. It does not need to be 3 times in a row ...
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0answers
11 views

Probability of runoff with a given return period

I've come across an issue, which I cannot figure out how to deal with in a statistical way. So, let's say I have $n$ events out of $N$ rain-events that produces runoff. Each of the $N$ rain-events ...
0
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0answers
32 views

How can I use mathematics to determine if my data is accurate?

Imagine I have an application which logs data about interactions in a shop. For example, whenever a customer comes in and purchases an item, the shopkeeper logs what item was purchased and the ...
4
votes
4answers
65 views

If a group consists of five girls and five boys, what is the probability that all girls will end up on the same team?

$10$ kids are grouped into an A team with $5$ kids and a B team with five kids. If the group consists of five girls and five boys, what is the probability that all girls will end up on the same ...
0
votes
2answers
29 views

100 shoelaces, pick 2 random ends and tie them together, what is the probability that a loop is created?

The question is: There are 100 shoelaces in a box. You pick two random ends and tie them together. Either this results in a longer shoelace (if the two ends came from different pieces), or it ...
2
votes
1answer
65 views

to be 99% certain of making a profit? central limit theorem?

Let $X_i$ be the profit card $i$ makes when its sold. I let $S_n = X_1 + ... + X_n$ so total profit. I found the mean of $X$ to be $0.1$. and $E[X^2] = 25$ so variance $= 24.99$ Are these correct? ...
2
votes
2answers
25 views

Divergence of asymmetric not-simple random walk

Consider a (not simple) random walk $S_n = \sum_{k=0}^n X_k$ where X_k are i.i.d and the mean $\overline{X}<0$. Is there is simple proof or a reference showing $P( \lim \limits_{k \to \infty} S_k = ...
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0answers
20 views

PMF for sum of uniformly distributed random variables

Let $X_1$ and $X_2$ be independent integer valued random variables that both are uniformly distributed on {1, 2, . . . n}. What is the PMF for S := $X_1$ + $X_2$? What I have so far: P(S=$X_1$+$X_2$) ...
1
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2answers
80 views

Functions of random variables result, where does it come from

I have learned that if one has two random variables, say $X$ and $Y$ and if $Y=g(x)$, then we have that density of r.v. $Y$ is: $$f_Y(y) = f_X(g^{-1}(y))\left| \frac{d(g^{-1}(y))}{dx}\right|$$ This ...
0
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2answers
50 views

How can I prove that expectation of conditional random variable?

I know the following results are true. However, I forgot to prove them. Please let me know how to prove them. $$E(X)=E(E(X|Y))\tag1$$ $$P(X)=E(P(X|Y))\tag2$$ (1) \begin{align} ...
0
votes
2answers
27 views

How can we find the bounds of the following integrals?

How can we find the bounds of the following integrals? $$\int_{0}^{+\infty}\int_{0}^{+\infty}\phi(x,y)dxdy$$ where $\phi(x,y)=$ $\begin{cases} 1 \quad if \quad x=y>0 \\ 0 \quad otherwise ...
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votes
1answer
51 views

What is the probability of getting intial state (read details)? [on hold]

Alex, Bob and Charlie each have 5 different colored marbles in their bags(same 5 colors in each of those bags though). Alex randomly picks a marble from Bob's box and puts it into his bag. Then ...
2
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2answers
19 views

Deck of cards probability with exclusion

What is the probability that a hand of five cards has exactly one club or exactly one heart So my logic was to select all possibilities with one club + all possibilities with one heart - ...
0
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0answers
16 views

martingale betting probability

What would be the probability of reaching a total of $40$ at a 50-50game using the martingale system with $20$ and standardbet 0.5? is it just 50% ? or is this strategy better/worse in short term ...
1
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1answer
37 views

Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
3
votes
1answer
1k views

Bus arrival poisson paradox

I have a question about the waiting time paradox for poisson processes(in this case in terms of bus arrivals). Suppose I know that buses arrive with poisson distribution(lambda). I arrive at fixed ...
0
votes
1answer
44 views

Expected value of X and Y for a given problem [on hold]

A couple decides to have children until they get a girl, but they agree to stop with a maximum of 5 children even if they haven't gotten a girl. If X and Y denote the number of children and number of ...
0
votes
1answer
46 views

How can two sets that do not intersect, be a subset of one?

How can two sets that do not intersect, be a subset of one? $$ C_{1},C_{2} \subset l $$ $$ C_{1} \cap C_{2} = \emptyset $$ $$ C_{1} \subset C_{2} $$ Specifically I am looking at the theorem that ...
0
votes
3answers
38 views

Value of a that Minimizes $\mathbb{E}([aX-\frac{1}{a}]^2)$

Suppose X is a random variable with mean $\mu$ and variance $\sigma^{2}$. For what value of a, where a > 0, is $\mathbb{E}([aX-\frac{1}{a}]^2)$ minimized?
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2answers
30 views

Normal distribution, probability and modulus question [on hold]

Say $X$ is a random variable which is normally distributed with mean $0$ and variance $1$. How do I find $k$ such that $$\mathbb{P}(|X-k| < |X+k|) = 0.7$$
1
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2answers
40 views

Probability of Moving Counters into Bags, Using Factorials.

Bag P and bag Q each contain n counters, where n > 2. The counters are identical in shape and size, but colored either black or white. First, k counters (0 < k < n) are drawn at random ...
2
votes
1answer
21 views

Probability - Finding the Support of a Joint Transformation

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right. $$ ...
1
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0answers
23 views

Guessing Mathematical Probabilities by Tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as many tests as I want. What is the minimum number of tests required to be at least at 95% sure I have the ...
0
votes
2answers
37 views

Finding the Expected Value with a Random Constant

Suppose $X$ is a continuous random variable with PDF: $$\begin{cases} e^{-(x-c)}\ \ \text{when }x > c \\ 0\ \quad \quad\text{when}\ x \leq c \end{cases}$$ a. Find $\mathbb{E}(X)$ b. ...
0
votes
1answer
26 views

Given irreducible transition matrix $P$, why does the matrix $(P−I|\mathbb{1})(P−I|\mathbb{1})^T$ have full rank?

Given irreducible transition matrix $P$, why does the matrix $(P−I|\mathbb{1})(P−I|\mathbb{1})^T$ have full rank? I know this is partially due to the fact that since $P$ is irreducible, there exists ...
0
votes
2answers
20 views

Given a Poisson distribution, $2f(0) + f(2) = 2f(1)$, what is the mean of the distribution?

If for a Poisson distribution $2f(0) + f(2) = 2f(1)$, what is the mean of the distribution? I know that for X ~ POI($\lambda$), then the pdf for the random variable X is \begin{equation} ...
1
vote
0answers
26 views

pde of probability density function [on hold]

I have a question about the time derivative of the probability density function, assuming a density function,$\rho(S,T|S_t,t)$, if I calculate the derivative with respect to T, that is the Fokker ...
2
votes
3answers
44 views

Let $X\sim N(3,4)$. Find $\mathbb{P}(X<7)$, $\mathbb{P}(X \ge 9)$

Let $X \sim N(3,4)$. Find $\mathbb{P}(X\lt7)$, $\mathbb{P}(X \ge 9)$, and $\mathbb{P}(|x-3|\lt 2) $ Okay lets figure out the PDF. $\mu=3$, $\sigma=4$. $$f(X)= \frac{e^\left(\frac{-(x-\mu)^2}{2 ...