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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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What is the shortest way to calculate sigma of Binomial Distribution?

Lets say I have a coin with P(heads) = 1/2 and P(tails) = 1/2. I toss that coin 100 times. What is then the probability of getting less than 45 heads? Because I though it is a Bin(100, 0.5) I tried ...
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1answer
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Probability of Alice getting ice cream twice in a month given constraints

Here is a practice problem I attempted to solve: Alice gets ice cream twice a month. Each pair of dates during the month is equally likely, except that the she never gets ice cream twice a day for ...
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Difference Entropy Output and Input

In a ternary channel with same probabilities for all inputs. I have H(X) entrance entropy and H(Y) output entropy. What does the difference of these 2 Entropy tell me?
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1answer
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For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
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1answer
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Why does the summation of these indicator variables start from i<j?

I'm currently reading through the eighth edition of A First Course in PROBABILITY, by Sheldon Ross. The section I'm reading is "Momens of the number of events that occur", and I understand everything ...
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2answers
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Does the probability of occurrence of a number remain same in bit level

Say, a number x occurs with probability p. x's binary representation be ABCD. So, does each of A,B,C or D is set with probability p?
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Point is randomly selected from area, find the mean of abscissa.

my question sounds like this: Point is randomly selected from area, which is limited with parabola y = x2 and straight lines y = 0, x = 0.97, x = 2.78. Find this point the mean of abscissa. This ...
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Majority voting 9 Bits Error probability

i have 1 Bit that will send 9 times. Each bit has an error probabiltiy of 1/3. The received Bits will be interpreted with a majority voting. Do anyone have an idea how i do check the probability ...
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1answer
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Law of Total Probability and Bayes Theorem Dice Question

Part i) before was asking for the definitions of these two theorems in the title so I assume they are supposed to be applied here. Which theorem will help to answer each part?
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Let $X$ be the random variable having probability density…

Let $X$ be the random variable having probability density $f(x) = \begin{cases} c|x-1|, & 0 < x < 2 \\ 0, & \text{otherwise} \end{cases}$ How to find $F (x) = P (X\le x)$?
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2answers
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An 8x8 cube is painted and cut into 1x1 cubes. One of them is taken a rolled, with bottom being blank. Probability that entire 1x1 cube is unpainted?

Andy has a cube of edge length 8 cm. He paints the outside of the cube red and then divides the cube into smaller cubes, each of edge length 1 cm. Andy randomly chooses one of the unit cubes and rolls ...
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Probability and statistics: )Estimate the number of the distributor's customers likely to replace their phones some time during the third year.

Previous experience shows that customers keep their phones for an average of 29 months.Assume a standard deviation of 12 months. A)Estimate the number of the distributor's customers likely to ...
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1answer
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Finding the probability of success that maximizes the variance of independent trials

A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer the jth problem correctly with probability $...
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0answers
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Expectation of an inner product in an infinite dimensional Hilbert space

Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
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1answer
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Expected value of the number of different numbers drawn in 37 rounds of roulette?

I need help with this problem. What is the expected value of the number of different numbers drawn in 37 rounds of roulette? Is this possible to interpret as the number of records? So the expected ...
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Compose a strategy for a game and compute the possibility of winning

There is a game. Alice has $k$ identical keys, which can open boxes. Bob have many boxes, one of which contains a diamond. If Alice opens the box with the diamond, then she wins. Bob will ...
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1answer
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How to appear probabilities X and find expected value,standard deviation,variance of an event X, in maple?

Is there any way to appear probabilities of X and find expected value,derivation and variance with a command in maple ?
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How to calculate covariance of independent standard normally distributed RV's?

Lets say I have 3 RV's that are standard normally distributed and independent : x , y and z Let X = 4*x + 2*y + 3 and let Y = 5*y + 2*z + 4 How can then the Cov(X,Y) be calculated? This is what I ...
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1answer
19 views

Estimating the probability that a particular batch of ten will include at least a faulty phone [on hold]

A uk distributor is to import $1000$ mobile phones which will be sold with an insurance package for $£400$ each. They will be sent out to a number of shops in batches of $10$. The distributor ...
9
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1answer
56 views

Bet with NBA players [on hold]

Giannis of Milwaukee Bucks and Kyle Lowry of Toronto Raptors are having lunch together at a restaurant. Giannis claims that if their teams play against each other, Raptors will lose with probability ...
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0answers
9 views

Probability - Birthday Problem [duplicate]

What is the probability that at least 3 students in a class of 21 students were born on the same month? Can someone please help with this question? Thanks.
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1answer
22 views

Finding density function of random variable

Choose an uniformly distributed random variable $U$ on the unit interval $[0,1]$. Then, what is the probability density function of $Y= \ln(U+ 1)$? I know the density function is the derivative of ...
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1answer
16 views

Distinct random variables in sampling

Let $X_1,X_2,\dots$ be i.i.d. random variables with $X_1 \sim U[0,1]$. Throwing out a null set all the variables are distinct. Can anyone explain this second sentence? What does he mean with "...
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1answer
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Finding the conditional probability of an event without indication if the events are independent.

I'm studying an introductory statistics textbook and, unfortunately, it doesn't come with an answer key. The textbook gave this problem, which I've spending hours trying to figure out: "Roll two fair ...
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1answer
24 views

Coin toss and geometrical distribution

Can we prove that random variable $X$ that counts the number of coin tosses until the first head/tail appear is geometrically distributed? I cannot seem to find such a proof anywhere. Is it even ...
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1answer
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Two equally strong teams, one having the upper hand for a long time

On one of my courses (applications of probability theory) the lecturer mentioned an interesting theorem. It was something along the lines of two, equally strong teams playing against each other and ...
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Reference request for convergence question in probability and statistics

Recently I have asked question here on math.se Regression covergence No one answered it so I decided to try to solve it on my own, however I don't know where to start. I had undergraduate courses in ...
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0answers
10 views

Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
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1answer
19 views

Confusion about the definition of a graph property and it's relation to $G(n,p)$

We defined a graph property the following way: Let $\Omega_n$ be the set of all graphs $G = ([n], E)$ on $n$ nodes. Then, a graph property is a sequence $P = (P_n)_{n \in \mathbb N}$ with $P_n \...
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1answer
15 views

Let $Z \sim G(p), W \sim G(2p)$ be independent random variables so that $ P (W>Z-1) = \frac{3}{7}$. Calculate $p$.

Let $Z \sim G(p), W \sim G(2p)$ be independent random variables so that $ P (W>Z-1) = \frac{3}{7}$. Calculate $p$. I tried to solve it this way: $$ P (W>Z-1) = \sum_{k=1}^{\infty} P(W-k+1>0,...
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1answer
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probability of Conditional Expectation

Is there a general formula for this type of questions? Given data: $$X \sim \mathrm{Geo}(0.09)$$ $$Y|X=x \sim \mathrm{Geo}(1/x+1)$$ How do I calculate $\mathbb P(\mathbb E(Y|X)=3)$?
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If $S_n$ is Binomial $(n,p)$ then $\mathbb P(S_n=k)\approx \frac{(np)^k}{k!}e^{-np}$.

I was reading this post, and I have to admit that I was quite confused. The question was : If $S_n$ is a Binomial r.v. with parameter $(n,p)$ s.t. $n$ large, $p$ very small and $np$ not to big (for ...
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0answers
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Variance of the range for the exponential distribution

If $n$ random variables are independently distributed with an exponential distribution $f_X(x) =\lambda e^{-\lambda x} (x\geq 0)$, the range $R_n = \max(X_1,\cdots,X_n) - \min(X_1,\cdots,X_n)$ has the ...
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1answer
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Let A and B be independent events such that P(A) = P(B) and P( AuB) = 0.5. Then P(A) =?

I tried solving this using $P(A\cup B) = P(A) + P(B) -P(A \cap B). $ I obtained the quadratic equation $2x - x^2=0.5. $ When I solved the equation where $x= P(A),$ I got the root values as 2.414 and ...
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1answer
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let $X$,$Y$ be independent Poisson distributed random variables with parameter $\alpha$ and $\beta$ respectively. $E(XY)$?

$$E(XY) = \sum_{x,y} xy f(x,y),$$ but I don´t have the $f(x,y)$. $X+Y$ would be Poisson distributed with parameter $\alpha + \beta$, but what about $XY$? Not sure what else to do. Thanks in advance.
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Strong Markov property and another stopping time

I'm trying to prove that given a regular continuous time Markov chain $X_t$ (pure jump process), its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous Markov chain, where $T_n$ is the time of the ...
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0answers
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Markov Chain holding time

Let $X$ be a continuous-time Markov chain. How does one justify $P(X(s)=x,0\leq s\leq t\mid X(0)=x)=\lim_{n\to\infty}P(X(kt/n)=x,k=0,1,\dots,n\mid X(0)=x)$ without prior knowledge of the ...
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1answer
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Calculate probability of union of events

Let $A$ and $B$ be events in sample space $S$ such that $P(A) = \frac{1}{2}$ and $P(A' \cap B') = \frac{1}{3}$. Find $ P(A\cup B')$. I found that $P(A\cup B) = \frac{2}{3}$ and $P(A' \cap B)$ [that ...
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0answers
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Is entropy function lipschitz? [on hold]

Is the information entropy function Lipschitz with respect to $l_{1}$ norm or $l_{2}$ norm?
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Probability that 5 students seated in a row are reseated such that none have the previous nieghbours

There are five students $S_1, S_2, S_3, S_4 , S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4, R_5$ arranged in a row initially $S_i$ sits on $R_i$. For $j = 1, 2, 3, 4$ ...
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Tricky information inequality $I(X;Z) \geq H(T)$

I am wondering whether $I(X; Z) \geq H(T)$ when the following conditions hold: $H(T | X) = H(T)$ $H(T | Y) = H(T)$ $H(T | X, Y) = 0$ $H(Y | Z) = H(T | Z) = 0$ $X, Y, Z, T$ are discrete. I know first ...
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1answer
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Probability drawing balls out of an urn

Seven balls are randomly withdrawn from an urn that contains $12$ red, $16$ blue, and $18$ green balls. Find the probability that $3$ red, $2$ blue, and $2$ green balls are withdrawn. The answer is $...
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2answers
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Counting sample points dice experiment

Suppose that die have been altered so that the faces are $1,2,3,4,5,5$. If the die is tossed five times, what is the probability that the numbers recorded are $1,2,3,4,$ and $5$ in any order? This ...
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0answers
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Hard demonstration - brilliant minds. I came up with the idea of ​obtaining the limit distribution of an estimator [on hold]

I do not know exactly how to get there, I thought to use the estimate of maximum likelihood but I'm not sure... If $X_1,\ldots , X_n$ are i.i.d. according to the uniform distribution ${\cal U} (0, \...
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0answers
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difficult demonstration, Convergence in distribution of subsamples? [on hold]

I am trying to solve this convergence but I am not know exactly what is the n1, please give me some ideas: Let $X_1$,...$X_n$ i.i.d with cdf $F(\varepsilon_p)$=$p$ and $f(\varepsilon_p)>0 $ if $\...
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2answers
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What is the average number of matches when randomly picking letters

Suppose we have three pieces of paper. On the first one you have the letter A, on the second on the letter B, and on the third one the letter C. Now suppose I'm going to randomly pick each one from a ...
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2answers
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Mean stopping time of a Brownian motion

I came across the following proof of the fact that the mean stopping time of a Brownian motion to hit $-1$ or $1$ is $1$: Let $B$ be a Brownian motion. We already know $B_t^2-t$ is a martingale. Let $...
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1answer
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Applying the definition of mutual independence to outcomes of random variables

My book defines mutual independence as: events ${A_1, A_2, ..A_n}$ are mutually independent if for any subset ${A_1, A_2, ..A_m}$ (where $m \leq n$) of these events we have: $$P(A_1 \cap A_2 \cap ......
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1answer
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Geometric Probability. Sphere

Suppose a sphere with radius $R$. Find the probability of the event such that $n$ selected points of the sphere are within the distance of $r = \frac{R}{2}$ from the center of the sphere. The points ...
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21 views

probability of picking same ball in n trials

Here is my problem: I have t unique hidden balls in an urn I must pick k balls each draw each time I draw a ball I can uncover it. I have n tries; in each try(draw) I have all t balls available ...