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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1
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1answer
19 views

When is $\nabla^2 f (x, y, z)= $ probability measure?

When is $\nabla^2 f(x, y, z) $= probability density function ? That is $\nabla^2 f(x, y, z)= \mu (x, y, z)$ $\int \mu (x, y, z) dxdydy = 1 $ What conditions must $f(x,y,z) $ satisfy? It is known to ...
2
votes
3answers
54 views

Probability of alternating sequence from uniform distribution

Say I sample a discrete uniform distribution $U$ (say $U$ has a support of $N$ elements, and there is a total order on the elements) a number $K$ times, resulting in a random sequence $A_{i}$. What ...
1
vote
1answer
44 views

Distribution of a process dependent on a Markov chain's states

Consider a Markov chain $X_t$ with state space $\{0,1\}$, initial distribution $$ \begin{array}{l} \mathbf{P}(X_0=1)=\lambda \\ \mathbf{P}(X_0=0)=1-\lambda \end{array} $$ and transition ...
1
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0answers
13 views

Diophantine Equation & Probability [on hold]

I'm asking please if there is any link betwen Probability and Diophantine equation ? For exemple can we estimate the numbers of solutions of the Diophantine equation: $F(x)=G(y)$ Thinks
-4
votes
1answer
39 views
0
votes
1answer
53 views

Distribution of max number of common balls

I have $n$ different balls numbered $\{1,\dots, i, \dots, n\}$. I choose $n$ balls, uniformly at random, with replacement. Let $X_i$ denote the number of times ball $i$ has been chosen. I would like ...
1
vote
1answer
32 views

Question on proof of Basu's theorem?

I've been trying to follow this proof on Basu's theorem-http://sankhya.isical.ac.in/search//64a3/64a3032.pdf . But Im getting stuck. The proof is on page 511. $T$ is a boundedly complete statistic and ...
1
vote
1answer
20 views

Expected sum value of permutaion

We have a set(A) of N elements. Let's assume elements are e1,e2,e3..etc. Value of each element can be 0 or 1. Another set of N elements(set B) are given, ...
16
votes
8answers
1k views

Probability that the sum of 6 dice rolls is even

Question: 6 unbiased dice are tossed together. What is the probability that the sum of all the dice is an even number? I think the answer would be 50%, purely by intuition. However, not sure if ...
1
vote
2answers
40 views

The inclusion and exclusion criteria

I've learned that in probability course, in the exercise we are asked to prove that: given $n$ sets $A_1,\ldots,A_n$, $$ \left|\bigcup_i A_i\right| \ge \sum_i|A_i| - \sum_{i\ne j}|A_i\cap A_j|\;.$$ ...
0
votes
0answers
30 views

Odds of summation of ten dice roll [duplicate]

If I flip 10 dice, what's the probability I get an Odd sum?? I couldn't do anything with this any help would be really appreciated...
0
votes
1answer
26 views

Example of submartingale $Z_n$ such that $Z_n^4$ is a supermartingale

Can one show an example of a submartingale $Z_n$ such that $Z_n^4$ is a supermartingale? I know that if $f$ is convex non-decreasing function then $f(Z_n)$ is also a submartingale. However, ...
0
votes
0answers
33 views

Probability of $\frac{P(X=k|X+Y=n)}{P(X+Y)}$

I wanted to ask about the conditional probability $\frac{P(X=k|X+Y=n)}{P(X+Y)}$. Having seen this a few times I know it becomes $\frac{P(X=k|Y=n-k)}{P(X+Y)}$. If X and Y are independent, why is the ...
0
votes
2answers
27 views

What's the best guess for the parameter of an exponentially distributed sample?

I have a sample of size $N$ values. I know the values are exponentially distributed, i.e. they are distributed according to this probability density function: $$ f(x;\lambda) = \begin{cases} \lambda ...
1
vote
1answer
21 views

What does it mean to say an ordered pair of vectors of random variables is independent of one anotehr

Suppose we are given i.i.d. random observations $\{ y_i,{\bf x_i} \}$ where $y$ is scalar and $\bf x$ is a vector. When one say $\{ y_i,{\bf x_i} \}$ and $\{ y_j,{\bf x_j} \}$ are independent of one ...
0
votes
0answers
23 views

Statistical method used to compare internet speed

A cable provider received a complaint from a customer. The customer is complaining that the internet service is too low. The cable company servicer measured the internet speed at two locations in the ...
0
votes
2answers
26 views

Probability of rejection (Sampling)

Consider a plant manufacturing chips of which 10% are expected to be defective. The chips are packed 30 to a box for distribution. A sample of size 10 is drawn without replacement from each box. If ...
-2
votes
1answer
23 views

Probability in normal distribution [on hold]

Random variable X has a bivariate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$. $$X = {x_1 \choose x_2}, \mu = {-2 \choose 7}, \Sigma = \begin{pmatrix} 3& -1\\ ...
1
vote
1answer
27 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.
1
vote
0answers
17 views

Relation between the expected number of visits to a state and reachability in a Markov chain

Let's consider a discrete time Markov chain $X_n$. Let $R_{ij} = \sum_{n=0}^\infty \mathbb{1}_{\{X_n= j | X_0 = i\}}$ be the number of visits to $j$ starting from $i$, and let $f_{ij}$ be ...
0
votes
2answers
22 views

Clarification on variance and expectation of Y value

Let $X$ be a random variable where $P(X = 1) = \frac13$ $P(X = 2) = \frac13$ $P(X = 3) = \frac13$ Let $Y = (X − 1)(X − 2)$ be another random variable that depends on $X$. What is $E(Y)$? I ...
0
votes
2answers
15 views

Probability of drawing either a heart, diamond or ace having removed some cards

just a quick question; Suppose that we take a standard pack of 52 playing cards and then remove the ace of clubs, the ace of hearts, the queen of hearts and the queen of diamonds. Suppose now that we ...
0
votes
0answers
58 views

On Casino Holdem game

For the game Casino Holdem, I am told to compute the expected value of the call option for the following hole cards and flop. (8♠,9♣) and (10♣,J♣,2♠) Can anyone help me on this problem. The rule ...
1
vote
1answer
19 views

Is there an interpretation of the hyper skewness?

Let $X$ be a random variable. The standardized $n$th moment of $X$ is defined as $$\frac{E[(X-\mathbb{E}[X])^n]}{\mbox{Var}[X]^{n/2}}. $$ Special cases are the skewness ($k=3$) and the kurtosis $k=4$. ...
0
votes
0answers
11 views

Combination: Selecting at least two cards, at least one from two non-exclusive sets.

I'm trying to figure out probabilities of certain hands for a game I've been considering. An example is the probability of having a card that is an Ace or King and a card that is a Heart or a ...
2
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4answers
59 views

I can't find or think of an intuitive way to think about P(A∩B) for non-independent events A and B.

I get that P(A∩B) = P(A) * P(B) for independent events. That makes complete intuitive sense for me. But I am struggling to think about P(A∩B) for events that overlap (that is, when B happening ...
0
votes
1answer
26 views

Independent and joint probability?

I got this question from my statistics teacher, but his answer made me confused. The question is this.. Given that A, B and C are three independent events such that P (A)=0.2 ,P(B)=0.6 ,P (C)=0.5, ...
-2
votes
0answers
19 views

I need help for questions c, d,e,f.g (Teaxas Holdem case study) formuls needed [on hold]

I need help with this because the answer submitted to my professor were wrong. a. The probability that you are dealt pocket aces is 1/221, or 0.00452 to three significant digits. Verify that ...
1
vote
2answers
98 views

Definition of a random variable $\mathrm{Var}(X)$

So $\mathrm{Var}(X) = \mathrm{E}((X-\mu)^2)$, but how can you subtract a function $(X)$ by a value ($\mu)$? And does it make sense to square a function?
2
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1answer
43 views

Composing dice throw probabilities

Suppose we are given a series of probabilities $p_a=0.2, p_b=0.1, p_c=0.5$ and $p_d=0.3$, for obtaining the value $4$ in a fair-dice throw. But the estimates were obtained for varying number of ...
0
votes
1answer
60 views

Expected value and sum of independent variables.

EDIT: I've found my mistake. Flipped around the values because in my head I had them tails up at the start.. Not sure what to do with the question now... On a table there are three coins in a row, ...
2
votes
0answers
13 views

Changing domain of the CGF of a stochastic process

Given a stochastic cadlag process $(X_t)_{t\geq 0}$. Define $A_{t}$ as the domain of the cumulant generating function by $f_t(s):=\log E(e^{sX_t})$, for which this expression is welldefined. I search ...
3
votes
1answer
594 views

Recurrence Relation, Discrete Math problem(Homework)

There is a disk, separated into n sections, as indicated in the graph. For each section, you can paint it with one color out of four: Red, Yellow, Blue, Green. The rule is adjacent sections can't have ...
0
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0answers
10 views

Probability of winning the NHL draft lottery from a subjective point of view

On April 30, the National Hockey League held its annual draft lottery to determine who would pick first in the 2016 entry draft. The 14 teams which missed the playoffs were eligible, and three teams ...
-1
votes
2answers
55 views

What does P(X=Y) mean?

Let X and Y be binary random variables, with $P(X = 0) = 1/4$, $P(Y = 0) = 1/4$ and $P(X = Y) = 1/2$ I want to calculate $P(X=x,Y=y)$ (i.e. probability of x and y) and P(X=x|Y=y) for all all x and y. ...
1
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0answers
25 views

Frog leap game - error anaylsis

We have a frog that finds itself on a complex graph. The vertices are labeled by integer indices, and the objective is to compute the probability for the frog being able to save itself, which ...
0
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0answers
16 views

Mathematical algorithm to assign robustness rating to Betfair Exchange Odds

The question I am asking also requires a bit of odds knowledge. Basically I need to come up with an algorithm/method that will assign a robustness rating to the Back And Lay Prices for each horse on ...
2
votes
1answer
24 views

Distribution Function is absolutely continuous or singular?

$$F(x) = \begin{cases} 0,& x < -1\\ \frac{x}{3} + \frac13,& -1 \leq x \leq 0\\ \frac{x}{3} + \frac23,& 0 < x \leq 1\\ 1,& 1 \leq x \end{cases}$$ This $F(x)$ is a distribution ...
2
votes
0answers
18 views

Deriving $(n-1)\dfrac{S^2}{\sigma^2} \sim \chi^2(n-1)$ [duplicate]

I can accept the fact that $Z^2 = \dfrac{\left(X-\mu\right)^2}{\sigma^2} \sim \chi^2(1)$ without knowing too much about this mysterious $\chi$-function, but I'm wondering how I can show that ...
0
votes
1answer
25 views

probability of splitted exponentially distributed random variables

Let $X$ be a exponentially distributed random variable(time interval) with mean of $u$ And $Y$ be a exponentially distributed random variable (time interval)with mean of $\lambda$ And $s$ be a ...
1
vote
1answer
793 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
1
vote
1answer
16 views

Expected number of trials until completion [duplicate]

You've got a discrete uniform distribution - what is the expected number of trials until each point is hit at least once? I started my thinking with maybe a Geometric distribution representing each ...
1
vote
1answer
30 views

Inequality for integral of complex valued functions

assume that $f$ is a complex-valued function acting on some probability space $(X,m)$ and $g$ a non-negative function defined on a same space such that $$ \lvert \int_A f \, dm \rvert \le \int_A g \, ...
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votes
0answers
25 views

What does it mean when e.g. P(3,4)? [on hold]

What does the following represent/mean: P(3,4)? With P standing for probability.
0
votes
0answers
15 views

Statistical Method used for industry remote test [on hold]

A manufacturer is producing special machine remotes. Once a month he is testing whether his production line is working properly. In order to do so the manufacturer is testing 100 remotes. The test is ...
0
votes
0answers
11 views

The probability of a book being borrowed is 20%. 2 books from a stack of 8 went missing. What is the probability the 2 books will be returned. [on hold]

The probability of a book being borrowed is 20%. 2 books from a stack of 8 went missing. What is the probability the 2 books will be returned
1
vote
0answers
31 views

How to prove that cardinality Borel $\sigma$-algebra equals the cardinality of $\mathbb R$?

My understanding at this point is that to assign a probability measure to a random variable defined on the real line, we need a Borel $\mathscr{B}$ sigma algebra, because otherwise we wouldn't be able ...
0
votes
1answer
36 views

Why $[0,1)$, i.e. “closed/open” in defining Borel sigma algebra?

The Borel sigma algebra over the interval $\Omega=\color{red}{[}0,1\color{blue}{)}$ can be defined as the sigma algebra generated by all open subintervals $\mathcal C_o$, i.e. $\sigma(\mathcal C)$, ...
2
votes
1answer
25 views

What is the probability that the sum of two dice rolls is a multiple of $3$?

What is the probability that the sum of $2$ dice rolls is a multiple of $3$? What about for $3$ dice rolls? For $n$ dice rolls? So I have the first part of this solution worked out by writing out all ...
0
votes
1answer
20 views

How to find Probability of Three People

Erik has .94 chance of being chosen, Bailey has .85 chance and Bert has .8 chance. What's the probability Bert would be chosen and Erik and Bailey would loose? What's the probability at least one will ...