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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7 views

Probability that sum of two uniformly distributed random variables is less than some constant

I am trying to find a way of determining the probability that the sum of two continuously uniformly distributed random variables is less than some constant $C$, formally: Let $A \sim ...
-2
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1answer
9 views

5 People roll a dice and flip a coin

Each of 5 people flip a coin and roll a dice (six sides). I know the total number of possibilities equates to $6 \times 2$ because the dice has 6 options, and the coin has 2 options. As a result we ...
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0answers
3 views

To calculate type I error of hypothesis testing on a discrete random variable

Suppose X is a random variable with $P(X=k)=(1-p)^kp$ for $k\in{0,1,2,...}$ and some $p\in(0,1)$. For the hypothesis testing problem $H_0:p=1/2$ and $H_1:p\neq 1/2$. Consider the test "Reject $H_0$ ...
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1answer
9 views

How does a change of measure affect covariance?

Suppose I have the three random variables $X,Y,M$ where $E[M] = 1$ under the measure $P$. Now, suppose I define a new measure $\widetilde P$ so that $\widetilde E[X] = E[M X]$ and $\widetilde E[Y] = ...
-1
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0answers
11 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
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0answers
12 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
2
votes
1answer
23 views

Conditional probability of 2 red marbles being selected

Suppose a bag exactly 5 marbles that are either red or green, and the probability of the bag containing 0, 1, 2,...,5 red marbles is uniform (e.g., each has probability 1/6). One person draws a marble ...
2
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0answers
20 views

Almost surely divergent series

Let $(X_{n})$ be a seqence of iid random variables such that $X_{n} \sim \mathcal{N}(\xi, \sigma^2)$, $\xi > 0$. I want to prove that $\sum_{i=1}^{\infty}\frac{X_{i}}{i} = \infty$ almost surely. I ...
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1answer
13 views

How do I determine the proper probability for a chance-based reward system?

Let's say I have game of chance with a number of players (1000), with each player having a chance to win something (\$25 for this example.) What probability to win would each player need in order to ...
2
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2answers
33 views

n distinguishable balls into n boxes

We have n distinguishable balls (say they have different labels or colours). If these balls are dropped at random in n boxes, what is the probability that: 1- No box is empty? 2- Exactly one box is ...
2
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0answers
32 views

Is this a Hilbert space? If not, is it reflexive?

Let $E$ be a Banach space. Let $L^2(\Omega, E)$ denote the space of random variables taking values in $E$ with second order moment. Is $L^2(\Omega,E)$ a Hilbert space? or at least, reflexive? 1) I do ...
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1answer
21 views

Find conditional probability of random variables

I need to find conditional probability to count mutual information. Random variable X has uniform distribution on set ...
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1answer
22 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
2
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0answers
16 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
0
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1answer
27 views

Question about unions of probabilities of disjoint sets.

Consider a probability space $(\Omega, A, P)$ and assume that the various sets mentioned below are all in A. Show that if $D_i$ are disjoint and $P(C|D_i) = p$ independently of $i$, then ...
1
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1answer
27 views

Any mathematical relation between these conditional probabilities

Let's say $A$, $B$ and $C$ are three different events. Is there any mathematical relation between these conditional probabilities: $\Pr(A\mid B,C)$, $\Pr(A\mid B)$ and $\Pr(A\mid C)$? Note: In the ...
1
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1answer
15 views

Intuition for partial averaging eqution

I just learnt about the condition expectation and as is known, the definition is as follows: My question is for the second property (partial averaging property), what kind of intuition does it ...
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0answers
18 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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1answer
30 views

Probability of $k$ collisions

Say we have $m$ buckets. We select a random bucket and put a ball in it, we repeat this $n$ times. In the end what is the probability of having at least one bucket with exactly $k$ balls? I have ...
1
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1answer
18 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
2
votes
2answers
42 views

divide 6 people in group of 2 in same size

Exercise: divide 6 people in group of 2 in same size. My solution: The exercise tells us to calculate the combination without repetition. If I start by calculating the number of ways to select how ...
2
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0answers
19 views

Updated conditional probability - Elo rating

Consider a tournament where all teams play against each other, and suppose we have the probability for win-draw-loss for every match (e.g. with the prediction from Elo rating). Is there a feasible ...
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1answer
24 views

probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
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0answers
15 views

Numerically stable way to compute the conditional covariance matrix

The Wikipedia article on multivariate normal distribution contains the well-known fact about the conditional "sub-distribution": If $μ$ and $Σ$ are partitioned as follows: $$ \boldsymbol\mu = ...
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0answers
34 views

Prove that the probability of getting at least $k$ heads when using coins that skew more to heads cannot be worse that the alternative

Let $0 \le p < p' \le 1$. Let $X_i$ be the Bernoulli random variable that takes the value of $1$ with probability of $p$ and zero otherwise. Similarly, let $X'_i$ be the Bernoulli random variable ...
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0answers
15 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
35 views

Length of pieces of stick broken at random [on hold]

A stick of length 1 is broken at random. How much longer is the longer piece than the shorter piece, on average?
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1answer
21 views

Probability of drawing balls from an urn with variable composition

A coin is tossed $k$ times, with probability $p$ of heads. In an urn, as many white balls are introduced as the amount of heads obtained, and as many black balls are introduced as the amount of tails ...
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1answer
25 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
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2answers
41 views

Jacod Protter “Probability Essentials” Problem 2.8

The question asks to show that a sigma-algebra $\mathcal A$ consisting of $A$ s.t. $A=f^{-1}(B)$, where $B$ is in $\mathcal B$ are Borel subsets of $R$ and $f$ is continuous, is contained in $\mathcal ...
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1answer
27 views

Probability question please guide

There are $5$ red and $3$ blue chips in a bowl. The red ones are numbered $1,2,3,4,5$ and the blue ones as $1,2,3$ respectively. if $2$ chips are drawn without repacement, find the probability that ...
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0answers
29 views

Determine the distribution [on hold]

Determine with justification the distribution (with parameters) of chimney fires in a large city over a week Assume the number of chimney fires over a year is 520.
2
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2answers
56 views

Is it true that E [ X | E [ X | Y] ] = Ex [ X | Y] ? Does this law have a name?

Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true: E [ X | E [ X | Y ] ] = E [ X | Y] Note that E [ X | Y ] is a ...
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0answers
14 views

Proving criterion for a transient state in Markov Chain

Let $\{X_n\}_n$ be a homogenous Markov chain. Prove that if exist a connected subset of states (means set of states which exist positive probability to move between them), $S$ which is not closed, ...
0
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1answer
28 views

What's the probability of the game being cancelled due to players not showing up

There are two teams, and each team has 6 players. 4 players are required for the game to go on. The probability of a player not showing up is $10\%$. What's the probability of the game being ...
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2answers
54 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ My current reading material claims ...
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1answer
19 views

Probability generating function and a discrete random variable

A discrete random variable $X$ has probability generating function $G_X(t)$. If $Y=aX+b$ show that the probability generating function of $Y$ is given by $G_X(t)=t^bG_X(t^a)$. Hence prove that ...
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3answers
2k views

On average, how many times must I roll a dice until I get a 6?

On average, how many times must I roll a dice until I get a 6? I got this question from a book called Fifty Challenging Problems in Probability. The answer is 6, and I understand the solution the ...
2
votes
1answer
33 views

$Y_n = \sup_{k \geq n} E(X_k | F_n)$ is a martingale if $X_n$ is $L^1$ bounded non-negative submartingale

Let $X_n$ be a $L^1$ bounded non-negative submartingale. Let $Y_n = \sup_{k \geq n} E(X_k | F_n)$. Show that (1) $Y_n$ is a martingale (2) $X_n \leq Y_n$ for all $n$ a.s. (3) $\sup \|X_n\|_1 = ...
2
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0answers
22 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
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0answers
13 views

discrete values probabilities problem [on hold]

Can you help me with this?? An engineer is requested to design water supply and waste water removal systems in a new industrial park consisting of 5 independent buildings. Assume that the water ...
3
votes
0answers
52 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
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0answers
20 views

Expected time until beating an initial try

Consider the following problem: Let $X,X_1,X_2,...$ be i.i.d. random variables. We execute the following experiment. One samples $X$. Then, one samples $X_1$,$X_2$ and so on until the first time the ...
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0answers
10 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
0
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1answer
41 views

A Question on CDFs and PDFs (substitution/inverse?)

(a) So there has been an answer to the question. Can someone explain how the limits of integration were found? I don't know why the upper limit is going to $X$.
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4answers
53 views

roulette with infinite money

Assumed I have infinite money and bet 10 on red or black and every time I lose I will double my bet until I win and then start again with 10 would I make profit? I did a bit code for that: ...
2
votes
2answers
18 views

Weighted Average Proof

Been stuck on this for a while now, seems pretty straightforward but can't seem to prove it. Given $\mu$ is a weighted average of $\mu_1$ and $\mu_2$ such that $\mu = x_1\mu_1 + x_2\mu_2$ where $x_1$ ...
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votes
2answers
20 views

Probability that two items selected from a mixed bag will be of particular sorts [on hold]

You have bought a mixed case of soft drinks. It contains six bottles of Coke, four bottles of lemonade, two bottles of tonic water, and eight bottles of mineral water. What is the ...
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0answers
16 views

Suppose that there are two cells in a parallel system. In order for the system to work, at least one of the two parallel subsystems must work.

Consider a particular lifetime value $t_0$, and suppose we want to determine the probability that the system lifetime exceeds $t_0$. Let $A_i$ denote the event that the lifetime of cell $i$ exceeds. ...
0
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0answers
9 views

sum of two Gaussian random variables conditioned on their sum

I have two independent standard normal R.V.s X and Y, and their sum is Z = X + Y. I am trying to calculate the PDF of X conditioned on Z taking the value z. I know that this is the joint PDF of X and ...