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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1answer
15 views

Is this equality holds? $\overline{F^{*2}}(x)=\int_0^x\overline{F}(x-y)dF(y)$

$X_1,X_2$ are non-negative i.i.d random variables with CDF F(x). I have a problem proving that following identity holds. $$ \frac{\overline{F^{*2}}(x)}{\overline{F}(x)}=1+\int_0^x\frac{\overline{F}(...
1
vote
1answer
789 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 \...
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0answers
8 views

Probability - Poisson Arrival Process

Car arrive at a toll booth according to the Poisson process at a rate of 3 arrivals per minute. a) What is the probability that the third car arrives within 3 minutes of the first car? b) Of the ...
2
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1answer
19 views

Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
0
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2answers
383 views

Conditional statestic

I have this exercise. A batch of 500 containers for frozen orange juice contains ten that are defective. Two are selected, at random, without replacement from the batch. (a) What is the ...
1
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3answers
37 views

I am stucked in some of this question about sample space and probability

What is the size of the sample space for the following scenario: Roll $3$ six-sided dice, and discard the highest roll In this question I know that sample space is all possible outcomes so the ...
2
votes
1answer
34 views

Conditional expectation of a product XY given Z with Y independent of Z

Let $X,Y$ and $Z$ be integrable random variables s.t. $XY$ is integrable and $Y$ is independent of $Z$ . I was wondering if there are any helpful/common ways of rewriting $\mathbb{E}[XY\mid ...
0
votes
1answer
391 views

Powers of Irreducible Transition and Periodic Transition Matrices

Suppose P is irreducible transition matrix with period d. How many communicating classes does P^k have and what is the period of each state?
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3answers
55 views

Probability of rolling 1-8 using six-sided dice

If I roll two six-sided dice where the first die is valued simply 1-6, but the second die is valued as 1-2=0, 3-4=1, and 5-6=2, and I total the two dice, will the probability of the numbers 1-8 be ...
0
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2answers
23 views

Query on Uniform random variable MAX & MIN

Llet $U$ and $V$ be independent, continuous uniform random variables on the interval $\left[1,5\right]$. Find $$\Pr\left(\min\left(U,V\right)<2 \mid \max\left(U,V\right)>2\right)$$
0
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1answer
25 views

For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely

Question in the title: For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....
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2answers
25 views

Probability of selecting same factor.

Willie Pikette randomly selects a factor of $144$. Betty Wheel selects a factor of $88$. What is the probability that they selected the same number? This is my incorrect approach (and please feel ...
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2answers
36 views

Bayes theorem and conditional probability [on hold]

I have a problem like this: Seventy-eight percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, ...
1
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0answers
22 views

Problems on continous random variable, probability, estimates.

Problem image: answering to questions (i) and (ii) i found that: $$pdf: f(x) = x \frac{2}{\theta^2}$$ $$E(x) = \frac{2}{3} \theta$$ $$Var(x) = \frac{1}{18}\theta^2$$ And now I tried to answer ...
1
vote
1answer
42 views

B-spline derivative

I used this control point vector: {{$0, 0$}, {$\pi/2$, $1.7$}, {$\pi$, $0$}, {($3\pi/2$, $-1.7$}, {$2\pi$, $0$}}, and this knots vector (by chord length method):{$0, 0, 0, 0.125, 0.25, 0.5, 0.75, 1, 1,...
25
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2answers
463 views

How long does it take a person with this “cheating” data-gathering strategy to achieve a desired result?

I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, I will cheat. Whenever I fail to have enough evidence, I will ...
1
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0answers
18 views

A result about weighted-sum of uniform random variables

Let $a_1,\ldots,a_m \in \mathbb{Z}$ and $U_1,\ldots,U_m$ be independent uniform random variables taking valules in $[0,1]^d$. Let $\mathcal{Z}$ be the support of the random variable $\sum_{i=1}^m a_i ...
1
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2answers
16 views

Evenly filling spaces for a specific average value

Imagine I have $N$ spaces. Each space can be empty, or occupied. Given a fixed point value $x$ between zero and one, I would like to evenly populate the $N$ spaces such that $\frac{N_{\text{occupied}...
0
votes
1answer
43 views

Two length 3 straights vs. one length 5 straight. Which is more likely and by how much?

Using a well shuffled standard $52$ card deck, $2$ players (call them A and B) decide to play a game. They draw community (shared) cards (without replacement) until a winner for that hand is ...
1
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0answers
21 views

Is there a way to maximize this probability by taking the derivative of the cumulative normal distribution function?

I'm self-studying Brownian motion and encountered the following problem. I understand the author's solution, and it is clear why maximizing the right-hand side of the inequality provides such $t$ ...
0
votes
1answer
387 views

Probability Question - Pls help

A salesperson visits $k$ clients each day. The salesperson makes one and exactly one sale each day. The probability that the salesperson makes a sale to the $j$th client on a given day is $p_j$, ...
0
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0answers
12 views

Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want to ...
0
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3answers
48 views

Odds of 10 thrown dice landing all the same

What are the odds of throwing $10$ six sided dice and landing all the same number. Also, how many throws would I need to do to achieve a $100$ percent success of this happening. Is this even possible ...
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0answers
16 views

Predict the daily usage of Bandwidth of a Network

Context: I want to predict the daily usage of bandwidth of a network (consists a number of users) based on previous use . For example, I want to predict the amount of bandwidth during 8 pm to 9pm ...
1
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1answer
31 views

How to find a mean probability

A speaks truth in 75 cases just of hundred while B speaks truth in 80 cases out of hundred.Find the number of cases where they are likely to contradict. I did try working it out.So out of 200 ...
0
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0answers
22 views

Detailed explanation needed for basic query regarding expectation

I need to find the expectation of following random variable $$g=[\log_2(\frac{1+x}{1+y})]^+$$ where $[x]^+=max(x,0)$ and both $x,y$ variables depend on variable $z$. I know the conditional pdf's and ...
0
votes
1answer
32 views

How can I generate a sample from the distribution $P(x) = \frac{exp(-(x^2-\mu)^2)}{\sum_{\bar{x} \in \mathbb{R}}exp(-(\bar{x}^2-\mu)^2)}$

I wish to generate samples from generate a sample from the distribution $$P(x) = \frac{\exp(-(x^2-\mu)^2)}{\sum_{\bar{x} \in \mathbb{R}}\exp(-(\bar{x}^2-\mu)^2)}$$. The unnormalized probability is $\...
3
votes
2answers
43 views

How do I prove that for a random variable $X$, we have $P(X \le a) \le p$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$, $\mu = \mathbb{E}(X) = 2$, and $\max(X) \le 10$, (or $P(X \ge 10) = 0$). How can I prove the following? $$P(...
0
votes
1answer
18 views

maximum possible probability

75% of the customers of ACME Mutual Insurance have auto insurance, and 40% have homeowners insurance. What is the maximum possible probability that a randomly selected customer with auto insurance ...
0
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0answers
40 views

How can I prove that for a random variable $X$, we have $P(X \le \mu) = P(X \ge \mu)$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$ and $\mu = \mathbb{E}(X) = 2$. How can I prove the following? $$P(X \ge \mu) = P(X \le \mu)$$ It is also ...
0
votes
1answer
14 views

how to get a distribution using the moment generating function

we have that X has a normal distribution with mean μ and variance 4. and we have to get the distribution of $(x-μ)^2/4$. I tried this: Y=$(x-μ)^2/4$, then $M_{y}(t)=M_{(x-μ)^2/4}=e^{μ^2t}M_{(x^2-2xμ-...
1
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1answer
22 views

Conditional expectation and variance of exponential distributions

Okay, so here's two problems from my book; Problem 1) Let $f(x,y) = 2e^{-(x+2y)}$ $x,y>0$ Calculate $V[Y|X>3 \cap Y>3]$ Solution since the joint density can be factored out into terms ...
2
votes
1answer
57 views

probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
1
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1answer
16 views

Chi-Squared Distribution

Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. $$ \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ...
0
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1answer
386 views

Conditional Probability and life expectancy

In a population of 100,000 females, 89.835% can expect to live to age 60, while 57.062% can expect to live to age 80. Given that a woman is 60, what is the probability that she lives to age 80? Using ...
2
votes
1answer
32 views

Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
2
votes
0answers
33 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
1
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2answers
24 views

Calculate probabilies based on given probability distribution

A mail-order company business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table \begin{array}{r|...
6
votes
1answer
95 views

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
0
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1answer
496 views

Figuring out probability of two random events both happening

So here's the problem: The table below shows the distribution of education level attained by US residents based on data collected during the 2010 American Community Survey: ...
1
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0answers
88 views

Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
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0answers
27 views

How to Calculate the “Drop Off” of a Set

So I have never taken a formal class of statistics and this is likely just a case of me not knowing the right name for what I am looking for. Nonetheless, say I have a set of numbers in descending ...
0
votes
1answer
718 views

How to calculate t-value, given degrees of freedom and $\alpha$.

While solving problems, we can look up physical t-tables or use a statistical analysis software like R to calculate t-values. But how do we actually calculate these values ? What is the algorithm ...
0
votes
0answers
24 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
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0answers
42 views

The Spacing of $e$ and $\pi$ Segments Within the Decimal Expansion of $\pi$

I discovered something seemingly very improbable today when I was searching for segments of $e$ and $\pi$ within the decimal expansion of $\pi$. I searched for $314159265$ and found it starts at the ...
2
votes
3answers
235 views

What are the odds of flipping a coin 100 times and seeing HHHHT? [on hold]

What are the odds of flipping a coin 100 times and seeing exactly four consecutive heads? Any more than four heads in a row, such as "HHHHH" would not be considered a string of four consecutive heads. ...
-3
votes
1answer
29 views

Conditional probability using set notation [on hold]

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
2
votes
0answers
62 views

When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
-2
votes
1answer
22 views

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). [on hold]

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). Solution is (1-z)^2
1
vote
1answer
29 views

Probability Sum

A purchasing agent must decide to accept or reject an incoming shipment of machine parts. The agent wishes to do either of the following: a1: Accept the shipment a2: Reject the shipment The fraction ...