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Questions tagged [probability]

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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

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$Y = \frac { K A ^ { 3 } } { ( B + D ) ( C - D ) }$

K is a constant Find an expression to approximately determine the variance of Y, assuming $A , B , C ,$ and $D$ are probabilistically independent. isnt the expression they have already given me the ...
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0answers
8 views

compare Bayesian linear regression vs standard linear regression

1st question, I recently learnt bayesian linear regression, but I'm confused that in what situation we should use bayesian linear regression, and when to use standard linear regression? What is the ...
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0answers
28 views

Ultimate extinction probability

I have the PGF: $$ G_x(\theta) = \frac{a-1}{a-\theta^2}\;\; a>1 = \text{constant} $$ I am trying to find the ultimate extinction probability $e$ and I think I need to solve $$e=G_x(e).$$ ...
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0answers
15 views

Stacked Conditional densities

I am struggling on the following problem (altered for better understanding): Let's say the height of a father $x_F$ is drawn from a normal distribution $X_{F} \sim \mathcal{N}(\mu,\sigma_F^2)$ and the ...
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0answers
24 views

Expected value of unique path between a and b

Let T be a full binary tree with 8 leaves. (A full binary tree has every level full). Suppose that two leaves a and b of T are chosen uniformly and independently at random. The expected value of the ...
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0answers
13 views

probability given standard deviation and mean using excel norm.dist function

I don't know if this is the right sub for this ,but I can find the probability something is less than i.e. NORMDIST $( \mathrm { x } ,$ mean, standard deviation, cumulative) but what if I want to find ...
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1answer
44 views

Markov Chain positive recurrence [on hold]

Consider a Markov Chain with an infinite state space $ \{ \dots, -1, 0, 1 , \dots \}$, where the transition function is defined as follows. On each step, we sample a function $f_{i_{n+1}}(x)$ from $$ ...
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2answers
144 views

finding the probability of an event, not equiprobable cases.

A person can be born on a Monday with probability 1/3 and on any other day with equal probability. What is the probability that 4 randomly chosen persons were born on different days of the week. <...
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0answers
28 views

Convolution of two step functions

Consider the probability distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $\lambda\equiv (\lambda_1,...,\...
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1answer
8 views

Decomposition of Joint Probability on Set

When we have two random variables $X$ and $Y$ with joint density $f_{X,Y}(x,y)$, we can find the CDF by basically integrating on all the values up to $x$ and $y$ for each variable. Now, I'm wondering ...
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0answers
35 views

A person with an ability to predict any future event with 30 percent accuracy makes a prediction of whether a coin toss comes up heads or tails.

If a person with an ability to predict any future event with 30 percent accuracy makes a prediction about the results of a coin toss. Does his prediction come out with more accuracy than a normal ...
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1answer
32 views

What will be the answer ??

I tried but was not able to find the correct way to solve this, I think this question is out of syllabus but still I want to know its answer
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1answer
35 views

Prove X and Y both have binomial distributions

Let $X$ and $Y$ be two independent, non-negative integer-valued, random variables and the distribution has the property $$ \Pr \{X=x|X+Y=x+y\}=\frac{{m \choose x}{n \choose y}}{{m+n \choose x+y}} $$ ...
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0answers
11 views

Exponential moment of return times markov process

For a discrete time irreducible + aperiodic markov chain (with general state space X) the geometric ergodicity criteria implies exponential return times to compact sets. More preciesly: The geometric ...
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2answers
23 views

Conditional Probability: All cards in a $5$ card hand are $\geq 8$, given that the hand contains at least one face card $(J, Q,$ or $K)$

This one has left me a bit frustrated. I think there are $^{12}C_1 + ^{12}C_2 + ^{12}C_3 + ... + ^{12}C_5$ ways to get a $5$ card hand with at least one face card $(1585)$. There should be $^{28}C_5$...
2
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1answer
18 views

Submartingale characterization

is there a characterization of submartingales in terms of stopping times similar to the martingale case in A martingale characterization. Especially is the following true: If $X$ is an adapted ...
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0answers
30 views

Adding the sum of 6 Dice, how do you calculate the permutations?

By working out the different combinations for the sum being 6,7 and 8, (1, 6, and 21) I can see this is a diagonal row in pascals triangle but I can’t work out how these would be calculated without ...
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2answers
23 views

When X,Y are independent random variables we can use convolution to find the density of $X+Y$, can we do that for $X-Y$? [on hold]

When X,Y are independent random variables we can use convolution to find the density of $X+Y$, can we do that for $X-Y$? Is there an analogical case?
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1answer
31 views

Cardinality of sets in a random graph generated from block model

Let $n\in \mathbb{N}$ be given. Let us assume that the set of vertices is $V=[n]=\mathcal{C}^+ \cup \mathcal{C}^- \cup \mathcal{D}$, where the sets $\mathcal{C}^+$ and $\mathcal{C}^-$ stand for the ...
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1answer
29 views

Why are Markov processes are completely determined by their initial value?

I'm reading a probability theory book, which (slightly reworded) says the following: A Markov process is completely determined once we know $$P_{ij}^{n,n+1} = P\{X_{n + 1} = j \mid X_{n} = i\} $$...
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2answers
37 views

Promise of formal definition of conditional expectation: what is $E[X|Y=y]$ exactly?

There are many questions here related to this but I'm yet to see one that directly address this issue. The promise of a formal definition of conditional expectation is that with it we may have a well-...
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2answers
24 views

Probability of an event occurs at least three times before another event occurs

cars arrives according to a Poisson process with rate=2 per hour and trucks arrives according to a Poisson process with rate=1 per hour. They are independent. What is the probability that at least ...
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1answer
20 views

How to define events to use in Bayes' rule?

"In transmitting dot and dash signals, a communication system changes 1/4 of the dots to dashes and 1/3 of the dashes to dots. If 40% of the signals transmitted are dots and 60% are dashes, what is ...
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0answers
10 views

Is there a connection between regularly varying tail with exponent $\alpha$ and the tail function itself?

Assume we know that the tail function $\bar F$ is regularly varying at infinity with exponent $\alpha$... Is it then possible to give a reasonable lower and/or upper bound for the tail function? So ...
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0answers
31 views

Infinitely often zero probability implies finite maximum

"For a sequence of events $\{U_n\}_{n=1}^\infty$, let ${1}_{U_n}$ be and indicator for $U_n$. Let $N = \text{max}_{n \in \mathbb{N}}\{1_{U_n} = 1\}$. $\mathbb{P}(U_n \: i.o) = 0 \Longleftrightarrow \...
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1answer
28 views

Standard Deviation of the random number x [on hold]

How do I solve this problem with a ti-83 calculator?
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1answer
20 views

If $Y$ = sum of $X$, what does the distribution look like when $X$ is poisson?

If $X_i$ is poisson. I know that the mgf of $X_i$ is $e^{\lambda(e^t-1)}$. What would the distribution of $Y$ look like if $Y$ = the sum of all $X_i$? is it Poisson itself?
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1answer
30 views

Comparing two die rolls of n-sided dice

I'm practicing my coding and designed a simple game that compares two results of a die roll modified by different multipliers. I can easily make these calculations in my code and determine the winner, ...
2
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1answer
10 views

Tail bound of sum less than sum of tail bounds

Problem: Suppose we have a probability space $(\Omega, \mathcal{M}, P)$, a random variable $X$ on this space, and two nonnegative measurable functions $f,g:\mathbb{R}\to[0, \infty)$. Choose some $\...
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0answers
22 views

Expected value of a function of multiple random variables [on hold]

I was wondering what is the definition of the expected value of a function of two or more random variables? And how does one show it is consistent? So if you have a random variable $z = g(x,y)$ ...
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1answer
20 views

Variance of Product of Ind. Variables

Whats wrong with my approach to answer the following question? The number of customers arriving to a fast food restaurant between 7 am and 9 am has the Poisson distribution with mean 40. Suppose that ...
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4answers
33 views

1 in 4, two chances, only one needs to hit

Currently in disagreement with a friend over the probability of a specific situation. My friend believes that if you have a 1 in 4 chance of something happening, getting two opportunities at it makes ...
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0answers
24 views

can a Markov chain have a periodic and transient state?

I want to say that in a Markov chain it is not possible for there to exist a state that is both transient and periodic. Here are the definitions I am working with. Let $P$ denote the transition ...
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0answers
12 views

Weak convergence of $(n^{-1}T_{k,n})_{n \geq \lceil k/2 \rceil}$

After a busy day, Santa asked $2n$ of his coworkers for dinner. There were $k$ elves and $2n-k$ gnomes. Santa and his guests sat randomly at the round table. Let $T_{k,n}$ be the number of creatures ...
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2answers
25 views

Probability measure domain

On a measurable space $(\Omega, F)$, where $\Omega$ is a set of outcomes and $F$ is a $\sigma -$field, what exactly is the domain of a probability measure $P$? If it's a specific $\sigma -$field such ...
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1answer
15 views

Cumulative distribution function and Brownian motion

I am having troubles with one exercise. Your help will be great! Let be $\Phi$ the C.D.F of a standard gaussian random variable (i.e. mean = 0 and variance = 1) and $(B_t)_{t\geq0}$ a Brownian motion ...
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0answers
38 views

Markov Chain generated by iterations of functions

$X_n$ is a Markov Chain on (..,-2, -1, 0, 1,..) obtained by random iterations with functions $f_1(x)=x+2$, $f_2(x)=x−1$, $f_3(x)=0$. In each iteration step we choose function to iterate with equal ...
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1answer
21 views

Flip an unfair coin

An unfair coin has an probability of heads on a single flip $p=\frac 14$, the coin is flipped n times, and the probability of getting 2 heads is the same as the probability of getting 3 heads, what is ...
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2answers
25 views

What is the probability that a set of nine children will contain three or fewer girls? [on hold]

I can't decide if it is 4/9 because there is the possibility of there being 0, 1, 2, or 3, or if it is 25% using combinations. Thanks!
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1answer
28 views

What is the probability that I actually won the game

Question I have two blue dice, with which I play a game. If I throw a double six (i.e. if I get two six on both the dices) then I win the game. I separately throw a red die. If I get a one, then I ...
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1answer
23 views

Sufficient condition for a Markov chain to be Aperiodic

If I want to prove that a Markov chain is aperiodic, then if I can show that $P(X_{n+1}=i\mid X_n=i)\gt 0$ $ \forall i$. Then can I say that the chain is aperiodic?
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0answers
18 views

How to accordingly adjust the % of a & b if I manually adjust c,d,e [on hold]

Item Share Adjusted A 10% - B 20% - C 5% 7% D 35% 30% E 20% - F 10% - Sorry I have edited to give above table. So based on sales the share of 6 items is given above. ...
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1answer
15 views

Convergence in mean of a series of random variables

I'm stuck on the following proof: Let $(X_1,.. X_n)$ random variables so that $E(Xi) = \mu$, $V(Xi) = \sigma^2$ final for all $1 \leq i \leq n$. It is also given that for all $i \neq j$, $Cov(Xi, Xj) ...
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0answers
29 views

Independent, exponentially distributed variables probability

The expected lifetime of a RV of type 1 is 2 years and the expected lifetime of a RV of type 2 is 1 year. Both are exponentially distributed and independent. I was therefore wondering how to calculate ...
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0answers
51 views

What is the probability two independent exponentially distributed random variables being greater than one another? [on hold]

I have two exponentially distributed independent random variables A and B. The expectations of $A$ and $B$ are $2$ and $1$ respectively What is the probability that $B > A$?
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2answers
65 views

Is it true that $\lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))$?

Suppose $G$ is a group. $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d. random elements of $G$ satisfying the condition that $$\forall H \leq G, \qquad P(X_1 \in H) = \begin{cases} \frac{1}...
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2answers
18 views

Reading Binomial Tables

While reading a table of cumulative binomial probabilities, if I need to find the probability of, for example, exactly 4 successful events happening and all the rest failures occurring, how would I go ...
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0answers
19 views

probability that a random variable is greater than a limit in given ordering of random variables

I am currently working on a modified version of the classic greedy algorithm for the 0/1 knapsack problem. Suppose that one has $N$ items with given weights and profits that are iid random variables ...
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1answer
33 views

Definition of ergodicity and ergodic process

I am confused by the definitions of ergodicity in wikipedia, see formal definition here which says that a measure-preserving transformation $T$ is ergodic if for every event $E$, $T^{-1}(E) = E$ ...
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2answers
37 views

Question on two independent random variables under Poisson distribution [on hold]

$X$, $Y$ are independent random variables, $X$ ~ Poiss$(λ)$, $Y$ ~ Poiss$(μ)$. How to find: a) $P( X > 0 | X+Y )$ b) $E( X | X+Y) $ ?