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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2
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3answers
71 views

$X,Y$ are iid from distribution $F$, which is a continuous function, then is $P(X=Y)>0$?

Suppose $X,Y$ are iid random variables from a distribution function $F$, which is a continuous function. Then is it always true that $P(X=Y)=0$? For me, the answer is trivially YES. We have $\int_y ...
0
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0answers
20 views

Difference in real vs computer simulated probability distribution results.

If this is not the right place to ask this,please guide me. I had a thought about what if our basic laws are somehow flawed such that they work in the situations we have observed but not in some ...
1
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0answers
46 views

Question on the hitting probability

I am confused on the relation between some basic operations with sets and probability. Consider a set $$ A=B\cup C\cup D $$ with $B,C,D$ disjoint sets. Take a random set $S$ almost surely non ...
0
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0answers
21 views

Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?

Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
0
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1answer
11 views

Use of Bayes theorem in the Lovásk local lemma

Here's a line from the proof on Wiki I don't understand. $$\Pr(A\mid\bigwedge_{B\in S}\bar{B}) =\frac{\Pr(A\bigwedge_{B\in S_1}\bar{B} \mid \bigwedge_{B\in S_2}\bar{B})}{\Pr(\bigwedge_{B\in ...
1
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1answer
38 views

what is the CDF of $f(x)=\frac{3x^2}{2}$?

This is probably a dumb question but I just want to make sure. The pdf is $f(x)=\frac{3x^2}{2}$ if $-1 \leq 0 \leq 1$. The CDF is $F(x)=\frac{x^3}{2}$ but with what bounds? sorry if this is an easy ...
0
votes
1answer
83 views

Sending bits and parity bits over noisy channel

Consider a sender is trying to send three information bits $a_1$, $a_2$, and $a_3$ over a noisy channel with error probability $$p = 0.001$$ That is with probability $p$ each bit may be flipped ...
0
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1answer
16 views

Is the mixture of Exponential family distributions an Exponential family distribution too?

Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. ...
-1
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0answers
26 views

Proof $\lim\limits_{n\to\infty} \dfrac{|S_n|}{A_n^{1/2} (\log_2 A_n)^{1/p}(\log_2\log_2 A_n)^{(1+\delta)/p} } = 0$

Let $\{X_k\}$be a random variables sequence and $S_n=\sum_{k=1}^n X_k$. I have $$ \limsup\limits_{n\to\infty} \dfrac{|S_n|}{A_n^{1/2} (\log_2 A_n^2)^{1/p}(\log_2\log_2 A_n^2)^{(1+\delta)/p} } \le ...
0
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2answers
42 views

How to find the right way to solve a given problem?

We distribute 10 indistinguishable balls to 5 girls. All the distributions have equal probability Let X be the number of girls who get at least 1 ball I need to find $Pr(X=3)$ and ...
1
vote
0answers
54 views

The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
1
vote
0answers
25 views

Expected maximum of maxima

Let $F(x)$ denote some CDF, and $\{f_i\}_{i=0}^m$ be a set of random variables independently drawn from that distribution. I would like to compute $$ E\bigg[ \max\bigg\{ \max\bigg\{\{f_i\}_{i=0}^m ...
1
vote
2answers
20 views

normal approximation with continuity correction

a fair die is rolled 100 times. What is the probability that "6" appears more than 15 times? Use the normal approximation with continuity correction. I've found the mean to be $100/6$ or $50/3$ and ...
0
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0answers
32 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
0
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1answer
24 views

How to set the limit for interated integral of $f(x,y)$ over diagonally partitioned region

I would like to compute $$I = \int_{\mathcal{R}} f(x,y) d\mathcal{R}$$ $$ f(x,y) = \begin{cases} x^2, \quad 0 < x < y < \pi \\ y^2 , \quad 0 < y < x < \pi \end{cases}$$ ...
1
vote
1answer
15 views

coincidence of recurrent random processes with infinite expected periods

That subject might not be quite accurate, but let me clarify. At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, ...
1
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1answer
35 views

How many draws to have a 90% chance of getting the ace of spades?

You have a standard 52-card deck, and you want to take the minimum number of draws from a random/shuffled deck such that you have a 90% chance of drawing the ace of spades. How would you find the ...
0
votes
1answer
40 views

Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = ...
1
vote
1answer
17 views

Probability problem involving Bayes' Formula

Two brothers share a car. They each have n keys in their pocket. They try one key at random and discard it until they can get the right one to start the car. Brother $A$ has only $1$ compatible key in ...
0
votes
1answer
23 views

What is the probability that smallest number is $6$ and largest is $15$?

Five numbers are drawn without replacement from the numbers $\{1, 2, 3, \ldots, 20\}$. What is the probability that the smallest number is $6$ and the largest is $15$? I am studying for a stats ...
1
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1answer
37 views

A probability problem about a thief?

Imagine that a detective is 60 percent sure that Mr.X is the thief in an investigation. 2 days later, They find new information about the real thief. The real thief is left-handed. Mr.X is left-handed ...
0
votes
1answer
42 views

Differences of heads and tails in a fair coin

I'm very new to this so I would appreciate a detailed explanation. I wrote a very simple Matlab program that "flips a coin" (randi([1 2])) $n$ times. Every time I ...
1
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1answer
24 views

What is the probability Amy wins a lottery prize for correctly choosing 5, not six, numbers…

Here is the full question: What is the probability that Amy wins a lottery prize for correctly choosing 5, not six, numbers out of six integers chosen at random from the integers between 1 and 40 ...
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0answers
18 views

Arrival times of taxis is poisson process

You are waiting in line for a taxi. There are two people ahead of you. Taxis arrive in a Poisson process, at average rate of one every two minutes Let T$_2$=time until 2nd taxi arrives I'm trying to ...
3
votes
2answers
40 views

What is probability that out of the first half on N objects, none will be matched with their own label?

The problem: We have N (even) objects ordered $o_1 ... o_N$ , each having their own label. The labels are reassigned to the objects randomly. What is the probability that that neither of the first ...
0
votes
1answer
33 views

Calculating probability that 5 dice rolls will produce an average of 3.5

We are carrying out a dice game which involves 5 people rolling dice and passing buttons equal to the number rolled . We have to work out the probability that we will get an average of 3 .5 items ...
0
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1answer
33 views

How to calculate Probability of “Ranking”?

I took a course with other 10 students. After the final exam, the professor reported statistical result of scores to us: 1. The range of score: 46~98; 2. The average score: 73.3; 3 The S.D of score: ...
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1answer
39 views

Probability - draw balls

Assume there are 100 balls in the box, 50 are white and 50 are black. What is the probability that I draw 9 balls in which at most 4 are white (without replacement)? $(p(9 black)+p(1 white + 8 black) ...
0
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1answer
37 views

Passengers on a plane [closed]

There are 200 passengers on a plane. 130 Italians, 45 Brits and 25 Chinese. 3 are extracted for a prize. Whats the probability of 1 Italian, 1 Brit and 1 Chinese being chosen?
2
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1answer
22 views

Understanding derangement.

From the inclusion-exclusion principle we get that out of $N$ objects with one label each, there is a probability of $$\sum_{k=1}^N (-1)^{k+1}\frac{1}{k!}$$ that a random assignment of the $N$ labels ...
1
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1answer
37 views

How many students would have to take the exam to ensure with probability at least $.9$ that the class average would be within $5$ of $75$?

I'm having trouble solving this problem: From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean $75$. How many ...
1
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1answer
41 views

If $X = X_1+\dotsb+X_N$, and $N\sim\operatorname{Pois}(\lambda)$, then what is the distribution of $X$ given $N$?

I have a question that I'm really struggling with (below): It's hard to understand exactly what is the question actually states. does this mean the number of trials itself is a distribution with a ...
2
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3answers
30 views

What is the probability that Person A will be chosen last every time$?$

There are $14$ people in a pool of subjects. People are selected at random one at a time. All people are chosen and then Person A.(Person A is chosen last every time This trial occurs $3$ ...
1
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1answer
25 views

Prob: Observation coming from a specific continuous distribution

I have a two-staged random process: First, I draw a type, which can be either $\tilde f$ or $\tilde g$. The (unconditional) probability of drawing the former type is $P_F$. Then, these two types ...
1
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1answer
31 views

Probability that one normal (uncorrelated) variable is greater than another if the latter is positive

Assume that $X\sim N(0,\sigma_x^2)$, $Y\sim N(0,\sigma_y^2)$ and $X$ and $Y$ are uncorrelated. Can we solve analytically for $\mathbb P(X>Y |Y>0)$?
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0answers
43 views
+50

Integrating a probability density function that only depends on the norm

I have a probability density function $f$ on $\mathbb{R}^3$ which only depends on the norm of a vector (that is, it takes the same value for $x,y$ if their length is equal). Let me call a region of ...
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0answers
50 views

Particle in a box

Say I have a point particle located at the center of a box and imagine that I give it a velocity v in some direction. It will bounce back and forth in different directions maintaining the same speed ...
1
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1answer
69 views

Sum of probabilities is infinite

I'm stucked solving this problem: Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with exponential distribution and $\lambda=1$. Show that ...
0
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1answer
25 views

Finding specific probability from a given conditional probability

Three girls A,B,C are going to meet. C is never late. The probability that no one will late is $0.4$ If it is known that at least one of the girls will be late then Pr(B late) = ...
0
votes
2answers
39 views

Find the density function from a joint density function

I try to solve the following task and I don't know what the correct way to do is. Let $p\in(0,1)$ and $(X,Y)$ be a pair of random variables with distribution density function ...
0
votes
1answer
22 views

Probabilty function of random variable

2 balls chosen randomaly from a jug. The jug containts $8$ white balls, $4$ black and $2$ orange. For each black ball, we earn 2 dollars. For each white ball, we lose 1 dollar. And for orange ball, ...
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0answers
14 views

Checking the independence of compound events

Given there are $n$ compound events, how many pairs of events does one have to check? As I understand it, one would have to check every pair, every triple, ... every $m$-tuple. Thus: $\sum_{k=1}^n {n ...
0
votes
1answer
28 views

Finding the approximate probability of random variables that are normally distributed

Let $X_1,X_2,\ldots,X_n$ be independent random variables each having the standard normal distribution. Find (approximately) $P(80<\sum X_j^2<120)$, $j=1$ to $100$
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0answers
42 views

For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
0
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1answer
22 views

Finding the cdf and pdf for $Z$, the standardization of $X$

Let $X$ be a normal random variable with parameters $\mu\in\mathbb R$ and $\sigma^2>0$. Find the cdf and pdf for $Z$, the standardization of $X$. What approach should I take on this? I initially ...
0
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1answer
57 views

I am not given figures to answer this question. Whats the right approach?

Z is a random variable defined as the sum of N independent Bernoulli trials where the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a ...
0
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1answer
29 views

Normal distribution pdf function returns value >1?

I am using the function scipy.stats.norm.pdf() in the following way: >>> scipy.stats.norm(scale=0.00026) >>> scipy.stats.norm.pdf(0.0005) 241.48 ...
4
votes
1answer
40 views

How many times $n$ must you play a game in which you have a $1/N$ chance of winning to have a better than 50% chance of winning at least once

I am having difficulty approaching the above problem, and would like a hint. I tired doing an inclusion exclusion argument: Let $A_{i}$ be winning the game on the i'th try, then by inclusion exclusion ...
0
votes
2answers
34 views

Expectation and Variance of random variable [closed]

I roll a fair die four times. Let X be the number of different outcomes that I saw. (For example, if the die rolls are 5, 3, 6, 6, then X = 3 because the different outcomes are 5, 3, and 6). Find the ...
1
vote
1answer
29 views

Why does the given condition imply the following random variables are not independent?

Let $Y ∼ U[0, 1]$ be uniformly distributed in the interval $[0, 1]$. Define the random variables $X_1, X_2$ as $$X_1 = sin(2πY )$$ $$X_2 = cos(2πY )$$ Why does the fact that $X_1^2 + X_2^2 = 1$ imply ...