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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Variance and standard deviation of probability distribution

What is the variance of this probability distribution? Round your answer to 3 decimal places. X 1.02 2.04 3.07 4.11 P(X) 0.24 0.25 0.28 0.23 Could you please break down. I ...
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1answer
98 views

How to find conditional expectation E(X|Y,Z)?

Question: Random variable X, Y and Z are as follows: Y=X+a Z=X+b where a~normal(0, $\sigma_y^2$) and b~normal(0, $\sigma_z^2$) a, b and X are independent of each other. Then how to find ...
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42 views

How can I find the expected value of a random variable with terms that increase until infinity?

Here is the question A company buys a policy to insure its revenue in the event of major snow storms that shut down business. The policy pays nothing for the first such snowstorm of the year and ...
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1answer
60 views

Linear Regression with independent but non-identical noise

If I have this linear regression equation: $$y=X\beta+\epsilon $$ ($x$ and $\beta$ are vectors) The likelihood function can be written as $$L= \prod_{n=1}^N N(y_n ;x_n ,\beta ,\sigma^2)=(2\pi ...
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3answers
5k views

A die is rolled 5 times, what is the probability that there are exactly 2 fives?

A die is rolled 5 times. Determine the probability that there are exactly 2 fives. The answer should be a decimal.
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142 views

Discrete RV problem

A flight control system uses four independent computers working in parallel. At each critical step, the computers “vote” to determine the appropriate step. The probability that a computer will ask for ...
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1answer
58 views

Random distribution of colored balls into boxes.

This is an abstraction of a real problem I have: I have a large number of balls that are either Red or Blue ($n = 9*10^7$) and a bunch of containers ($c = 3*10^7$). I've calculated that the ...
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1answer
36 views

Let $T$ be exponential with parameter $\lambda$. Let $X$ be discrete defined by $X= k$ if $k \leq T < k+1$, $k=0,1,2,\dots$. Find the pdf of $X$.

To be honest, I am lost on this question. Here is what I have so far: $$ \ F_T(t)=- e^{-\lambda t}=P[T\le t] \ $$ $$ \ P[X=k]=P[k\le T \lt k+1] \ $$ I am not sure how to go about finding the pdf for ...
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1answer
89 views

Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population.

Recall that for a $N(\mu,\sigma^2)$ population $W=\frac{n-1}{\sigma^2}S^2\sim \chi^2(n-1)$. [a] Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population. ...
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1answer
50 views

Prove that $E[Y_1 + Y_2\mid X=x] $…

Let $Y_1,Y_2,X$ be random variables I want to prove that $E[Y_1 + Y_2 \mid X =x] = E[Y_1 \mid X= x] + E[Y_2 \mid X=x]$ I'll do it in the discrete case as follows; $$\sum_{y_1} \sum_{y_2} (y_1 + ...
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1answer
55 views

What to do when a probability problem becomes unwieldy to check via simulation?

I am assuming that some probability problems cannot be solved easily since there may be a lot of cases to handle and it would make miscounting likely. However, some problems do not simulate well on a ...
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1answer
14 views

Expected Homogenization Time

Assume we have $N$ boxes, and each box contains one red sock and one blue sock. We can then perform the following process: randomly take one sock from each box and replace it with a red sock. What is ...
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0answers
27 views

Convergence of random saddle point

Let $y_n^*$ be the solution of $$ y = g_n(y) $$ where $g_n(\cdot)$ is a random function. Suppose that for fixed $y$ $$ g_n(y)\to h(y) $$ almost surely and pointwise as $n\to\infty$. Is there any ...
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1answer
73 views

How do you calculate the probability of an observation falling within a certain percentile?

Say you have a very large dataset with a known mean and SD. Then, you take a very small number of observations from this dataset (say, 30). How do you calculate the probability of one of those ...
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2answers
158 views

Probability of touching a specific ball at least once in 100 tries from a box of 100 well mixed balls.

We have a box which contains 99 red balls and 1 green ball. The box is shaken vigorously such that each ball has an equal chance of being touched. A guy with covered eyes goes to touch one ball but ...
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1answer
57 views

Find the upper and lower limit of integration (marginal distribution)

I have a joint density of two random variables: $f_{Z,V}(z,v) = \dfrac{1}{\alpha}$, $\alpha\geq1$ .The question is to find the marginal distribution of $Z$. the area of integration is given by: ...
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2answers
105 views

Chevalier de Méré's Problem Type Question

Is the following argument correct: A double six in a single turn in game B is 1/6 as likely as rolling a six in one turn in game A. But there are 6 times as many turns in game B as game A. Thus the ...
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1answer
78 views

Confidence interval for the conversion on site

I am the developer of web service and I'm trying to to build some plots for the inner dashboard. I raised two questions that I can not solve on their own. Suppose ...
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1answer
36 views

Joint distribution of two gaussians, one of which is dependent on the other.

Suppose $x\sim N(\mu_x,\sigma_x^2)$ and we are given that $y\mid x \sim N(a+bx,\sigma^2)$, where $a$ and $b$ are some constants. It is a fact that the joint distribution of $x$ and $y$ is a bivariate ...
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3answers
564 views

Probably of 2 six in 5 dice rolls

What is the probability of obtaining exatcly 2 six when rolling a dice 5 times? In order to obtain this probability, I will need to devide the number of favorable events by the number of possible ...
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1answer
74 views

inverse Laplace transform by integral

I've seen this formula for the inverse Laplace transform in several books: $$f(t)=\mathcal{L}^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\alpha -iT}^{\alpha +iT}e^{st}F(s)ds$$ where $f$ is ...
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3answers
46 views

Find the $P(X>1)$ for the given pdf?

A part of this question asks me to find the $\Pr(X>1)$ given that $$f_X(x) = \begin{cases}\frac{1}{\sqrt{4x}} & 1 <x<4 \cr 0 & \text{otherwise} \end{cases}$$ I solved this by taking ...
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1answer
112 views

Gambler's ruin: Distribution of the maximum fortune along the game conditioned to lose

I having troubles with this problem: Let $(X_n)$ a gambler's ruin Markov chain on $\{0,\dots,N\}$ i.e. a Markov chain with state set $E=\{0,\dots,N\}$ and probability transitions $$p(k,k+1)= ...
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2answers
117 views

multiplication rule questions - 7 people attending a concert

7 people are attending a concert. (a) In how many different ways can they be seated in a row? (b) Two attendees are Alice and Bob. What is the probability that Alice sits next to Bob? (c) Bob ...
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0answers
44 views

Probability of a number of weighted items being allocated to the same bin

I have the following (probably classic) combinatorics problem: There are $n$ bins that can hold $k$ items each, and a total of $r = n\,k$ items. The items have weights $w_1 > w_2 > \dots ...
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1answer
134 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
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1answer
29 views

Prove that $F(N/2;N,x)+F(N/2;N,1-x)=1$ where $F$ is binomial CDF

I have the following claim: $$F(N/2;N,p)+F(N/2;N,1-p)=1$$ where $F$ is a binomial CDF with exactly $N/2$ successes in $N$ total trials, and with each trial having success probability $p$. Is it ...
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3answers
73 views

Non-continuous Distribution Function

Given $$F(X) = \begin{cases} 0 \text{ for }x < -1\\ \frac{x+5}{10} \text{ for } -1 \leq x < 3\\ 1 \text{ for }x \geq 3 \end{cases}$$ which is a discrete and continuous distribution function, I ...
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1answer
25 views

I this the correct approach?

I want to make sure I'm thinking about this question right. Ten students are traveling home from college. Amoung them they have two cars, each of which will hold six passengers. How many ways can they ...
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1answer
47 views

Probability Proof - Bayes law multiplicative rule

I have been trying to prove that P(A∪B|C) = P(A|C)+ P(B|C) - P(A∩B|C) I have gotten to P(C|A∪B)*[P(A) + P(B) - P(A∩B)] = P(C|A)*P(A)/P(C) + P(C|B)*P(B)/P(C) + P(C|A∩B)*P(A∩B)/P(C) but I am ...
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1answer
73 views

Roll fair die until you get (1)

Roll a fair dice until you get (1). Let X = # of rolls needed. Similarly, for a second die, Y = # of rolls needed to get (1). For M = min(X,Y) find the pmf and E(M). This is a sample problem, i have ...
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3answers
129 views

Random variable stochastic bigger than random variable

I have a exercise, which I don't know how to show. It goes like, X is a continuous random variable with support $(-\infty,\infty)$. Consider the random variable $Y=X+\Delta$, where $\Delta$ is ...
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1answer
155 views

Divide line segment into 2 parts

I have a problem with this exercise, which says: I have to divide a line segment into two parts, where I select a point randomly. I need to find the probability that the larger segment is at least 3 ...
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1answer
26 views

How to articulate this expression

I was walking a student through the binomial expansion process and remarked that I prefer Pascal's triangle to generate the coefficients. He also needed to know this way of producing the numbers. ...
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2answers
239 views

I don't understand why the solution to this probability question is set up in this manner

The problem states: A pair of dice is cast until either the sum of seven or eight appears. (a) Show that the probability of a seven before an eight is 6/11. (b) Next, this pair of dice is cast ...
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1answer
35 views

Probability theory problem , inequality

Let $n$ be a positive integer and $a , b$ two randomly chosen positive integers not exceeding $n$. Let $P(n)$ denote the probability that $\begin{equation} \\ a ^ 2 + b ^ 2\leq n^2 \end{equation}$. ...
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1answer
69 views

what is the probability that the contractor's estimate will be within 5 weeks of the true mean

A contractor uses sample mean lifetime $x'$ of $250$ compressors as her estimate for population mean lifetime m of all new compressors. If this brand of compressors has a standard deviation of $35$ ...
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2answers
51 views

Amount of ways to schedule activities using combination or permutation.

I'm trying to review for Probabilities and Statistics and came upon this Question. If one needs to schedule a job interview for someone who wants to teach at a school. For the day of the interview, I ...
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2answers
72 views

Marginalization question $\Pr[a] = E_X[\Pr[\ a|X\ ]\ ]$

I'm reading explanation of a theorem, and there's one step that I can't understand. I know it should be simple enough, but I just can't think of the reasoning atm. The step says, According to ...
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2answers
16 views

$L^1(P)$ random variable limit.

for random variable $X$ in $L^1(P)$, What is $\lim_{a\to \infty}a * P( |X| > a )$ ? I think its value equals 0. But I can't solve this problem. I used markov's inequality But I can not ...
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4answers
85 views

Rephrasing die question

Seems I asked my last questions poorly and it got a bad response. Okay so basically here's the problem and I will explain how I am trying to do it. You roll 1 fair die repeatedly until you either ...
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1answer
85 views

Probability Rolling a dice until you get 1/2 or 2/2 [closed]

You roll $1$ die repeatedly until you get a $1$ immediately followed by a $2$ or a $2$ immediately followed by a $2$. What is the probability that a $1$ then a $2$ is rolled before a $2$ then a $2$? ...
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1answer
289 views

Product of sequence of uniform integrable random variables are uniformly integrable?

If $\{X_i\}$ for $i \in I $ and $\{Y_j\}$ for $j \in J$ are uniformly integrable.Then prove that, $\{X_i+Y_j\}$ for $(i,j) \in I \times J$ is uniformly integrable.What about $ \{X_iY_j\} $ for $(i,j) ...
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1answer
60 views

Convergence in distribution and probability

Suppose ${X_{n}}$ is a sequence of non-negative random variables with cumulative distribution function given by $F_{X_{n}}(x) = 1 - 1/(1+nx)$ for $x\geq 0$. Examine if $\{X_{n}\}$ converges in ...
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1answer
40 views

Bound for the variance of a stochastic process

Given a random variable $X$ and $N$ realizations of the stochastic process associated to $X$, a theorem gives a bound for the $\sigma^2[X]$: $$\sigma^2[X]\le\frac{1}{4}(A-a)$$ where $A$ and $a$ are ...
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1answer
101 views

Chebyshev Inequality

I am reading a research paper, and the author claims to get to a desired result by making use of the Chebyshev Inequality. I can get to the desired result also with some reasoning, but I fail to ...
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0answers
40 views

If a sequence of events $A_n$ converges to 0, then does the probability of the intersection equal 0?

Here's an elementary question which I believe is true, but my intuition about infinite intersections is not very solid. Let $A_n$ be a sequence of events such that $P(A_n) \rightarrow 0$ as $n ...
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1answer
27 views

Rearranging Question

Hi guys I have come to the following inequality but can't seem to workout how to get to my final step. I know it is just rearranging the equation but I haven't been able to get it or I would like ...
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3answers
570 views

Probability that a student knows the answer

The Probality that a student knows the correct answer to a multiple choice question is 2/3 . If the student does not know the answer , then the student guesses the answer . The probality of the ...
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1answer
17 views

Finding the probability of a probability density function

Suppose that $f(x) = e^{−x}$ for $0 < x$. find $P(1 < X)$ I know typically we integrate $f(x)$ from $1$ to $\infty$ but in this case $x = 1$ is not included, how do I go about doing this? All ...