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
votes
3answers
948 views

What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6?

Similar to: "What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?" but we allow repeats so 1,1,2,2,3,4,4,4,4,5,5,6 would count. My answer (or simulation) is flawed as I ...
1
vote
2answers
924 views

How do I find the sampling distribution for variance from sampling distribution of $\bar{X}$?

Suppose some people are asked to each randomly pick 30 apples and put into a bag. In this case, the weights of all the bags will surely be different because different apples weigh differently. So ...
-2
votes
1answer
115 views

conditional distribution X_1|X_1+X_2=r

suppose $f(x_1,x_2)=p^2q^{x_2},\ x_1=0,1,\ldots,x_2,\ x_2=0,1,2\ldots$ how can find $\mathbb{Pr}(X_2-X_1\leq1)$? also if $(X_1,X_2,X_3)\sim M(n,P_1,P_2,P_3)$ find conditional distribution ...
6
votes
2answers
944 views

What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?

If I have a fair die and throw it until I get a run of 1,2,3,4,5,6 in order, how many times on average must I throw the dice?
0
votes
3answers
425 views

Find Probability that a player will win Nth match

Please help me with this question: Player "A" starts the first game. Player who starts a game has probability "P" of winning that game. Player who loses starts new game. Assuming this series ...
0
votes
1answer
104 views

What are the $X_1, X_2, …, X_n$ in the Sampling Distribution of $\bar{X}=\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ X_{ j } } $?

The sampling distribution of $\bar{X}$ defined in a book that I am reading is $\bar{X}=\frac { 1 }{ n } \sum _{ i=1 }^{ n }{ X_{ j } } $. I know $X_1, X_2, ..., X_n$ are random variables. But what is ...
1
vote
1answer
125 views

In search of memorable example of “(Pearson-)uncorrelated $\not\Rightarrow$ independent”

I am looking for an easy-to-remember (and non-trivial) example that vividly illustrates that the "uncorrelatedness" (in the sense of Pearson) of two random variables $X, Y$ does not imply that $X$ and ...
1
vote
0answers
178 views

Conditional independence

Need some help with this exercise (Exercise 4.1 Probability Theory, E.T.Jaynes): Suppose that we have vectors of events $\{H_1,...,H_n\}$ and $\{D_1,...,D_m\}$ which satisfy: (1) $P(H_i H_j)=0$ for ...
4
votes
2answers
215 views

The law of large numbers: How does the convergence take place? How does the “remainder” look like?

I have the independent and identically distributed random variables $X_1,X_2,\ldots$ with a finite expectation $\mu$. I also have defined $S_n = X_1 + \cdots + X_n$. According to the law of large ...
4
votes
3answers
148 views

Probability of finding 2012 before any other occurence of 012 in a random infinite sequence of digits 0,1,2

The following problem is from the semifinals of the Federation Francaise des Jeux Mathematiques: One draws randomly an infinite sequence with digits 0, 1 or 2. Afterwards, one reads it in the ...
10
votes
2answers
12k views

What is the probability that a solitaire game be winnable?

By "solitaire", let us mean Klondike solitaire of the form "Draw 3 cards, Re-Deal infinite". What is the probability that a solitaire game be winnable? Or equivalently, what is the number of ...
3
votes
0answers
145 views

Bayesian inference on partitioned multivariate Gaussian

My question is on Bayesian inference of partitioned multivariate Gaussian. To make things easier, suppose there is a 2-dimensional Guassian, $$ X_1 \sim N(\mu_1, \sigma^2_1) \\ X_2 \sim N(\mu_2, ...
1
vote
1answer
81 views

Is this question related to Poisson process?

Consider a computer system which employs two copies $A$ and $B$ of some chip. A chip $C$ on reserve is used to replace either $A$ or $B$ whichever fails first. What is the probability that $A$ is ...
0
votes
0answers
151 views

Gauss-Markov process and conditional density function.

$x(k+1)=A x(k)+G d(k)$ $x(k)$ Gauss–Markov process, d(k) white process with average $\overline{d}$. The conditional expectation of $x(k)$ is $E\left[x(k+1)|x(k)\right]=A x(k)+G \overline{d}\ \ \ \ \ ...
1
vote
2answers
114 views

How to solve without the binomial law?

A perfect dice is drawn $4$ times.What's the probability to the same number comes out at least $2$ times? At first, I applied to binomial law.I made all calculations. I set up a random ...
-1
votes
2answers
83 views

Integrability Properties with Norms

Let $\mu$ be a probability measure on $X \subseteq \mathbb{R}^m$. Prove that $|| a + B x ||^d$ is integrable iff $||x||^d$ is integrable: $$ \int_{X} || a + B x ||^d \mu(dx) < \infty \ ...
0
votes
2answers
90 views

How to expand $p(A|B)$ when $p(C)$, $p(A|C)$ and $p(B|C)$ is known?

This is a silly and basic question however I got myself confused. Suppose $A, B, C$ are r.v, how to expand $p(A|B)$ when $p(C)$, $p(A|C)$ and $p(B|C)$ is known? Does it hold $$p(A|B) = \int ...
2
votes
2answers
417 views

If conditional expectation $E[Y|X]$ is constant.

It is well-known fact, that if $X,Y$ are independent, integrable random variables then $E[Y|X]=E[Y]$. Next assume that $Y$ is centered and $E[Y|X]=0$. What reasonable conclusions can be made about the ...
0
votes
1answer
114 views

Independent random variables.

We define probability distributions $F(x,y)=P(X\leq x, Y\leq y)$ where X,Y are two arrays of random variables, and probability density function $\frac{\partial^{n+m} F(x,y)}{\partial x_{1}...\partial ...
2
votes
2answers
326 views

Card probability question

A 52-card deck is thoroughly shuffled and you are dealt a hand of 13 cards. (a) If you have one ace, what is the probability that you have a second ace? (b) If you have the ace of spades, what is the ...
3
votes
2answers
802 views

How to prove this property in a Poisson process?

For a Poisson process show, for $s < t$, that $P(N(s)=k|N(t)=n) = \binom{n}{k} (\frac{s}{t})^k (1-\frac{s}{t})^{n-k}$
0
votes
2answers
379 views

How to decide what is the probability distribution in a Monte-Carlo simulation?

For a Monte-Carlo integration of $$\int_\Omega P(x)f(x)\ \text d x,$$ there seems to be no apriori distinction if $f$ or $P$ is the probability function. So does it matter if I consider $$P, f, P ...
1
vote
1answer
138 views

Calculating probability, given any word of length N?

Suppose I have 2,096,896 randomly generated letters (they were actually derived from pi). How can I calculate the probability that a word of length N will appear? I took discrete math a few years ago, ...
1
vote
0answers
60 views

Sampled or discretised Gaussian Random Variables

Suppose I have $X$, a normal random variable with mean $\mu$ and variance $\sigma^2$. Now I discretise this random variable to form a discrete random variable $Y=g(X)$. $Y$ could be created by ...
0
votes
1answer
270 views

Density of truncated normal distribution?

I have a truncated normal distribution with mode $0$ and variance $\sigma^2$ that only consists of non negative values. What is the density of this distribution at some non negative $x$? I have just ...
2
votes
1answer
109 views

Designing an efficient sampling strategy

In a Monte Carlo simulation, my goal is to compute an estimate of the mean of a distribution via sampling. Traditional, straightforward statistics generates samples (via simulation) and computes the ...
1
vote
2answers
162 views

Expectation of the difference of sums

Let a be vector in $R^{2m}$. I would like to calculate $E|\sum_{k=1}^ma_{\pi(k)}-\sum_{k=m+1}^{2m}a_{\pi(k)}|^2,$ Here $\pi(\cdot)$ is a permutation on the set{1,...,2m} with uniform distribution. ...
1
vote
1answer
107 views

Poisson distribution question attempt

I usually find these sort of questions straightforward but this was worded pretty vaguely as I haven't really come across anything like it. I answered all the questions but I'm not sure about the ...
1
vote
1answer
808 views

Binomial distribution of number defective in a sample of 25 randomly selected

I'm having difficulty answering the essay" statistics questions I keep encountering in my practical work. Here are questions and answers in particular: A manufacturer buys many thousands of a ...
0
votes
2answers
81 views

Problem determining the probabilistic model

Is my probabilistic for the problem of determining the probability, that a arbitrary chosen boy has a sister, if it is equal likely that a family has boys or girl and if the probability of having ...
1
vote
1answer
3k views

Binomial distribution question

I'm doing some practical work on the binomial distribution but currently finding it difficult to answer iii-c. Here is the full question and the answers I've provided. A bank claims that 80% of ...
1
vote
4answers
437 views

probability and statistics

I really need help with this question. The coefficients $a,b,c$ of the quadratic equation $ax^2+bx+c=0$ are determined by throwing 3dice and reading off the value shown on the uppermost face of ...
0
votes
0answers
118 views

Linearity property of Mathematical objects

I can't help notice that linearity property is rather common among various mathematical objects like finite dimensional vector spaces and functionals, Expected Value of random variables. I am sure ...
1
vote
1answer
489 views

Gaussian Noise Covariance Matrix in the Extended Kalman Filter

In Simultaneous Localization and Mapping: Part I, the Extended Kalman Filter is described on page 5. I'm confused about where it says "$w_k$ are additive, zero mean uncorrelated Gaussian motion ...
-1
votes
1answer
127 views

expected value problem

Say we have a probability density function $f_y(y)=3y^2$, where $0\leq y\leq1$ and we take 15 observations at random. If $x$ is a number within the interval $(.5, 1)$ what is $E(x)$?
3
votes
1answer
308 views

Convergence in Probability $\Rightarrow$ Convergence in Expected Value

Under which conditions the Convergence in Probability implies the Convergence in Expected Value?
2
votes
1answer
783 views

Find probabilities of winning in a modified version of Monty Hall Problem

So I modified the original version of the Monty Hall problem and allow there to have 4 doors; 1 car and 3 goats behind the doors. I will choose one door, Monty, who knows where the car is, randomly ...
0
votes
4answers
281 views

How many permutations?

I'm trying to calculate the number of possible non-repeated permutations of these serial key styles. I have no mathematical background and cannot read formulas, which is why I'm struggling with ...
1
vote
1answer
351 views

Constructing the transition matrix for device failure

There are three machines. Let the probability that an operable machine fails on any given day be $0.1$, independently of the other machines. Only one machine can be repaired on the same day (so it is ...
2
votes
2answers
4k views

Probability that two cards are black and one is red

If I draw three cards at random (without replacement) from a standard 52-card deck, what is the probability that two of the cards will be black and one of them will be red? Thanks!
0
votes
3answers
792 views

How buying “back in” in poker tournament changes odds

My buddy and I are arguing over something that cropped up in this past weekend's Texas Hold'em tournament. A player got "knocked out" (lost all their chips) early on in the game. The person hosting ...
3
votes
2answers
2k views

Prove that the sample median is an unbiased estimator

My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
0
votes
1answer
58 views

Putting balls and dice in baskets

We have 20 objects, 10 of which are dice and 10 of which are balls - and we can distinguish them all - and we have two baskets; and in each basket we throw 10 objects. Can someone please explain to me ...
1
vote
1answer
106 views

Simplified Multinomial Distribution

I am working with a simple case of the multinomial distribution, as follows: There are $k = 8$ different possible outcomes, each occurring with equal probability $p = \frac{1}{8}$. What is the ...
2
votes
2answers
737 views

Distribution of sum of iid binary random variables

I have a sequence $X_i$ of random variables which can with probability $1/2$ each, take values $+1$ and $-1$. How do I find $\lim_{n \to \infty} P(\Sigma X_i \le x)$? Pretty obviously the sum has ...
0
votes
1answer
74 views

A problem involving probability mass shift

Consider a probability model with sample space on the interval $[0,a]$ where $a$ is a finite positive real number. Consider two probability distributions $P_1$ and $P_2$ on the sample space, where ...
0
votes
1answer
68 views

Pdf of a product

Suppose $X_{1}$ and $X_{2}$ and i.i.d random variables. Consider $K = X_{1}X_{2}$. Then does $f_{K}(k) = f_{X_{1}}(x_1) \cdot f_{X_{2}}(x_2)$?
0
votes
1answer
69 views

Independence of Random Variables II

Suppose $X$ and $Y$ are i.i.d. random variables. Also suppose they take the values from the set $\{1,2, \dots, n \}$. Then does this mean that $$P(X=1, Y= 1) = P(X=1) \cdot P(Y=1)$$ $$P(X=1, Y=2) = ...
1
vote
0answers
256 views

Independence of Random Variables

Suppose we have three random variables $X_1, X_2$ and $X_3$. For pairwise independence it is sufficient to show that $$E[X_{1}X_{2}] =E[X_1]E[X_2]$$, $$E[X_{1}X_{3}] = E[X_1]E[X_3]$$ and ...
0
votes
2answers
168 views

Find $P(X\gt Y)$ using the joint density

$f_{X,Y}(x,y) = \frac{2}{3} (x+2y)$ for $0 < x < 1, 0 < y < 1$; find $P(X\gt Y)$. I got 1/9 by evaluating $$\int_0^1\int_0^{x-1} \frac{2}{3}(x+2y) dy dx$$