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 views

How to approach a hypothesis test problem

I have a specific problem I'm working on, but I can simplify it to an example problem like this: I have two biased coins, one (A) which generates heads 10% of the time and one (B) which generates ...
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0answers
3 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
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0answers
11 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
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1answer
14 views

A problem on continuous random variables

I was reading a The First course on Probability by Sheldon Ross, while I stuck at this possibly stupid doubt. The problem is : The density function of X is given by $$ f(x) = \begin{cases} 2x, ...
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1answer
26 views

A roofing company has 8 roofing jobs to complete in the next two months in how many different orders can the roofing jobs be completed ??

A roofing company has 8 roofing jobs to complete in the next two months in how many different orders can the roofing jobs be completed ?? Please answer as soon as possible thanks so much !
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0answers
20 views

How to hedge a sports bet

Suppose I've got a $200 ticket on the Golden State Warriors to win the NBA Finals at 5 : 1. The finals start next week, with the Cavs listed at 2 : 1 to beat the Warriors and the Warriors 4 : 9 to ...
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2answers
18 views

Events With States - Random Walks

A mosquito is walking at random on the non-negative number line. She starts at $1$. When she is at $0$, she always takes a step $1$ unit to the right, but, from any positive position on the line, she ...
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1answer
27 views

Can anyone help clarifying the geometry in this probability, random variables question.

So basically the question is to find the CDF of $Z$ where $Z$ is the random variable that signifies the distance from a point in a square(sides 1 length) to a fixed vertex of the square. I do not ...
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1answer
14 views

Pdf of a normal variable accepted with probability dependant on the normal variable

Assume $z$ is a standard normal variable. If $z<0$, then we accept it with probability 1. if $z\ge0$, we accept it with probability $e^{-mz}$, where $m>0$. I'm trying to figure out the pdf of ...
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0answers
17 views

Random projection onto orthonormal bases

Given an arbitrary N dimensional vector of length $L$, and a $M$ dimensional orthonormal basis chosen uniformly at random with $M<N$, what is the CDF of the length of the projected vector?
2
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1answer
16 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...
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0answers
16 views

Substitutions in Probability Generating Functions

I have a probability generating function: $$G_{i,j}(x,y)=\sum_{i,j}p(i,j)x^i y^j.$$ I was wondering what is the intuition beyond setting $G(y)=G_{i,j}(1,y)$? Does it represent anything special? I ...
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1answer
41 views

Probability of getting extra heads

Suppose you are tossing a coin $10$ times. You observe that the first $5$ tosses result in all heads. Then what is the probability that you would get $3$ heads in the remaining $5$ tosses? I have no ...
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2answers
33 views

determing the probability distribution

I have 2 sets of elements, say $A=\{a\}$ (only 1 element) and $B = \{b_1, b_2,..., b_n\}$. The probability of picking $A$ is $0.3$ and the probability of picking $B$ is $0.7$, and all elements in $B$ ...
1
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1answer
12 views

Group testing of individuals for disease

The problem asks to find the expected number of tests required to find all the individuals who are infected with a disease. Occurence of the disease has a probability of $0.002$. There were two parts ...
2
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1answer
33 views

Random variable $X$ is given with the density function $ \phi_X (x)= \frac{1}{2} e^{-|x|}$ Find the distribution of the random variable $Y$ if:

$$Y=\begin{cases}-X-2,\ \ \ \ X \leq -1 \\ \ \ \ X, \ \ \ \ \ -1 \leq X \leq 1 \\ \ \ \ \ 1, \ \ \ \ \ \ \ \ \ X >1 \end{cases}$$ Now I'm only interested in $t >1.$ (That is only ...
1
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1answer
10 views

Continuous random variables and probability density function

OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called ...
1
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1answer
36 views

Probability of getting high heads total

Probability of getting high heads total in long coin-toss sequence If I flip a coin 300 times, what is the probability of getting head at least 200 times?
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0answers
17 views

Probability Question

On a fighter plane 3 single-round shots are fired. The probability of success for the first shot is 0.4, the second 0.5 and the third 0.7. It is sufficient to fire 3 shots in order to destroy the ...
0
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1answer
35 views

How many different arrangements that end in “LL” are possible using the letters from the word “COLLEGE” [on hold]

How many different arrangements can be made that end in "LL" are possible using letters from the word "COLLEGE" Please answer as soon as possible thanks !
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2answers
26 views

I am trying to calculate the standard deviation of sales demand.

198 days total. Here are the sales numbers for days with a sale(16 days): 24,20,10,10,10,2,6,10,12,10,12,12.24,1,13,12 182 days without no sales what is the standard deviation? should the 182 ...
2
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0answers
11 views

Consequence of random walk with positive speed on a graph

Consider a random walk $X(n)$ on a vertex-transitive graph where the random walk has positive speed, i.e., $$ \lim\limits_{n \rightarrow \infty} \frac{d(X(n), X(0))}{n}= \alpha>0$$ almost surely. ...
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1answer
27 views

Calculating probability of digital roots

I am trying to find correlations in words that share the same single digit digital root. I will assign a correlation if there is the same difference between the n digit digital roots of the words, or ...
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0answers
24 views

Entropy of $\operatorname{Beta}(\alpha, \beta, a, c)$

I know that the differential entropy of the two parameter Beta distribution $X \sim \operatorname{Beta}(\alpha, \beta)$ is $$ \begin{align} h(X) = \ln \operatorname{B} (\alpha, \beta) &- ...
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1answer
27 views

Distribution of the product of a Normal and an Exponential random variable

What is the probability distribution of $M$, given $M=V*X/k$, where $X$ is Normal, $V$ is Exponential, $k$ constant? Or, in the real world, the probability distribution of (Cost/k) where ...
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0answers
11 views

Distinct pairs formed using repeated sequences

Problem setting: Let $N$ and $M$ be positive integers. Let $I'$ and $J'$ be ordered sets, i.e., sequences, $\{1,2,\ldots,N\}$ and $\{1,2,\ldots,M\}$ , respectively. The sequences are $N$ and $M$ in ...
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1answer
15 views

find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts

Can you give me a breakdown of the stages you take arriving at the answer to the following question: Jan buys $5$ packets per week with a $30\%$ chance of finding a gift per packet,find the ...
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2answers
35 views

Poisson Distribution Greater than problem

A company manufactures long continuous lengths of computer network cable. The manufacturing process is not perfect, and sometimes faults are present in the cables. Faults occur along the cables ...
1
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1answer
29 views

Expected Value to grab a ball

Say we have $b$ blue balls and $r$ red balls in a urn. Randomly we grab a ball out of the urn, until we grab a blue ball. Now I want to find the expected value of the number of balls that have been ...
5
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1answer
67 views

Convergence of a sum of random variables

Let $(X_n)$ be a sum of i.i.d. positive random variables such that $\mathbb{E}(X_1)=1$ and $\mathbb{P}(X_1\neq 1)>0$. Put $M_n=X_1\ldots X_n$. Show that $\sum _{n\geq 1}\sqrt{M_n}< +\infty $ ...
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0answers
17 views

Kino 90 numbers. You choose 10 numbers. What is the % of getting your number [on hold]

Kino 90 numbers. You choose 10 numbers. Starting from 1... to 90. There is a (say) 1/3 chance of this number being drawn. Example. Number 1... roll a 3-sided dice. I roll a 3--the number is drawn. ...
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0answers
21 views

Mean and standard derivation

A set of numbers consists of one's and three's. Find the mean and standard derivation if there are 23 one's and 17 three's. What's the meaning of this question ? I know the formula for mean and s.d. ...
0
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1answer
20 views

Finding the mode of the negative binomial distribution

The negative binomial distribution is as follows: $f_X(k)=\binom{n-1}{k-1}p^k(1-p)^{n-k}.$ To find its mode, we want to find the $k$ with the highest probability. So we want to find $P(X=k-1)\leq ...
2
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1answer
37 views

Using a markov chain to calculate the expected value of conditional/constrained choices (TopCoder PancakeStack)

I've been working on a programming challenge (link) where an expected value is calculated. Recently I learned about Markov chains and successfully applied them to solving a set of problems, but the ...
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1answer
17 views

Calculating Percentages/Probabilities of a Specific Scenario.

I've been trying for a few hours now to wrap my brain around, what at first, seemed like a simple concept. I've tried so many different things and ways of constructing scenarios so I can figure out ...
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2answers
30 views

Proving the $Pr(d>0|a+d=\pi)$ is increasing in $\pi$ when a and d are two independent normal distributions.

I was wondering if it is possible to prove the following (or show false otherwise). Given two independently distributed random variables $a\sim \mathcal{N}(\alpha,\sigma_\alpha^2)$ $d\sim ...
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2answers
20 views

Independent Events find P(A or B')

Given that P(A and B)=0.1 and P(A and B')=0.4 find P(A or B') if A and B are independent. The ans is 0.9 Please tell me the rule you use in the problem.
1
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1answer
27 views

Distance From Point to Nearest Value in Series

Let's say I have a point, chosen at random from the range [0, 1]. What is the average distance of this point to the nearest point in a set of n points chosen at random from the same range? ...
1
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1answer
34 views

Average number of tries needed before success

there is a 3% chance of success there are a thousand people trying over and over until they succeed how many tries will it take on average for the last person to achieve this success? I know that ...
1
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0answers
22 views

Compute the stationary distribution of a Markov Chain on an infinite state space

I have a Markov Chain on $\mathbb N_0^2$ with a given initial state $(x_0,y_0)$. The allowed transitions for example are of the following form: $(x,y) \mapsto (x-1,y+2)$ with probability $\propto x$ ...
2
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1answer
44 views

Mathematics of contamination [on hold]

I want to know the distribution of residual material (contamination) in subsequent refills. For example, suppose a cup normally used for transferring salt is used, without cleaning, for transferring ...
1
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0answers
8 views

How to get an approximation of $P(A\leq \max_{1\leq i\leq n}B_i)$,where $A, B_i$ are independent Gaussian random variables

Consider the independent Gaussian random variables, $A$, $B_1$,...,$B_n$. $B_i$ is distributed as $N(0,1)$. They are all independent. $A$ is distributed as $N(m,1)$. How can I approximate the ...
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1answer
31 views

It is true that $\int_{0}^{\infty}\mathbb{P}(x<m \ \cap Y \leq k-x) f_{X}(x)dx= \int_{0}^{m}\mathbb{P}( Y \leq k-x) f_{X}(x)dx$?

Let $X$ and $Y$ be independent random variables. Then it is true that? $$\int_{0}^{\infty}\mathbb{P}(x<m \ \cap Y \leq k-x) f_{X}(x)dx= \int_{0}^{m}\mathbb{P}( Y \leq k-x) f_{X}(x)dx$$ And, how ...
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0answers
22 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
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2answers
32 views

Finding the mode of a distribution

I've been trying to get a better understanding of distributions. So far I understand how we get the formulas for mean and variance (by looking at the derivative of the moment generating function). ...
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1answer
35 views

Help needed with Probability Question [on hold]

A card is drawn at random from a deck of playing cards. If it is red, the player wins 1 dollar; if it is black, the player loses 2 dollars. Find the expected value of the game. I think that the ...
3
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3answers
858 views

Probability of drawing a pair of brown socks

You have a drawer with $6$ loose blue socks, and $10$ loose brown socks. If you grab two socks from the drawer in the dark (random draw), what is the probability that you draw a brown pair? I have ...
0
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1answer
44 views

Rolling 1 die 5 times [on hold]

One die is rolled five times. How many different results are possible? Of those, in how many ways can there be exactly 2 rolls of 4? For the first part I multiplied 6 five times and got 7776. I'm ...
2
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1answer
17 views

Conditional probability with max(X, Y)

Let $Y_n=$ the outcome of the $n$-th die roll, let $X_{n+1} = \max \{X_n, Y_{n+1}\}$ with $X_1=Y_1$. What is $P(X_{n+1}=j \ | X_1=i_1, ..., X_n=i)$? I know that it is $P(\max \{X_n, Y_{n+1} \}=j \ | ...
0
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0answers
17 views

multivariate interval estimation

I have several samples of probabilistic vectors, i.e, each sample is of the form $(x_1, \cdots, x_n)$ such that $\sum_{i=1}^n x_i\leq 1$ (they are sub-probabilistic vectors), how can I obtain a ...