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
votes
1answer
19 views

Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
1
vote
2answers
15 views

Calculate probabilies based on given probability distribution

A mail-order company business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table \begin{array}{r|...
0
votes
3answers
34 views

Odds of 10 thrown dice landing all the same

What are the odds of throwing $10$ six sided dice and landing all the same number. Also, how many throws would I need to do to achieve a $100$ percent success of this happening. Is this even possible ...
1
vote
0answers
17 views

How to Calculate the “Drop Off” of a Set

So I have never taken a formal class of statistics and this is likely just a case of me not knowing the right name for what I am looking for. Nonetheless, say I have a set of numbers in descending ...
1
vote
0answers
35 views

The Spacing of $e$ and $\pi$ Segments Within the Decimal Expansion of $\pi$

I discovered something seemingly very improbable today when I was searching for segments of $e$ and $\pi$ within the decimal expansion of $\pi$. I searched for $314159265$ and found it starts at the ...
-3
votes
1answer
25 views

Conditional probability using set notation [on hold]

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
2
votes
3answers
209 views

What are the odds of flipping a coin 100 times and seeing HHHHT? [on hold]

What are the odds of flipping a coin 100 times and seeing exactly four consecutive heads? Any more than four heads in a row, such as "HHHHH" would not be considered a string of four consecutive heads. ...
-2
votes
1answer
16 views

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). [on hold]

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). Solution is (1-z)^2
0
votes
0answers
20 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
0
votes
2answers
31 views

Find the limit of the probability of uniform random variable?

Let $X_1 ,X_2 ,X_3 ,…$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Then, calculate the value of $$\lim_{n\to \infty}P(-\ln(1-X_1)-\ln(1-X_2)-\cdots-\ln(1-X_n)\geq n)?$$ My work: Since ...
1
vote
0answers
16 views

Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
-1
votes
0answers
15 views

recursive definition for two mutually exclusive events [on hold]

How do we write recursive definitions for two mutually exclusive events ? Can anyone explain with some examples as how do we come up with solutions in case of exclusive events ? SO finally i add ...
1
vote
1answer
25 views

Probability Sum

A purchasing agent must decide to accept or reject an incoming shipment of machine parts. The agent wishes to do either of the following: a1: Accept the shipment a2: Reject the shipment The fraction ...
0
votes
0answers
20 views

Variance: logical/mathematical meaning [duplicate]

$$\operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right]$$ Is the formula of variance. But if you think of it, you can assume that square was introduced just that something other than ...
-4
votes
0answers
37 views

Length of left stick [on hold]

Break the stick into 3 pieces and what would be the expected length of left stick? I need answer for this to verify my answer. Can somebody give me the answer? My thoughts: My answer is 1/4. First ...
1
vote
0answers
23 views

Is there a known probability distribution with cumulative probability function $F(X)=\frac{1-X^a T_1}{1-X^a T_2}$? [on hold]

I have a random variable $X$, which is distributed with the cumulative probability function $F(X)=\frac{1-X^a T_1}{1-X^a T_2}$, where $a$ is negative. I am wondering is there any famous distributions ...
2
votes
1answer
22 views

Simplification of probability expression

Let $p_1$, $p_2$, and $p_3$ be probabilities such that $p_1 + p_2 + p_3 = 1$, and let $c_1$, $c_2$, and $c_3$ be arbitrary constants. Can the following expression be written in terms of $p_1$ and ...
0
votes
1answer
17 views

Bivariate Normal Distribution properties questions

$\newcommand{\Var}{\operatorname{Var}}\newcommand{\Cov}{\operatorname{Cov}}$we have that $X$ and $Y$ are random variables with a bivariate normal density with parameters $(μ_x,μ_y, σ^2_x, σ^2_y, ρ_{xy}...
1
vote
2answers
29 views

How to approach this conditional probability question

The problem is as follows: Let X be the winnings of a gambler. Let $p(i)=P(X=i)$ and suppose that $p(0)=1/3;\\ p(1)=p(-1)=13/55;\\p(2)=p(-2)=1/11;\\p(3)=p(-3)=1/165$. Compute the ...
1
vote
1answer
28 views

Moment generating function of $X+Y$ using convolution of $X$ and $Y$

Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$; show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X and Y$ are independent and continuous. I am confused ...
1
vote
0answers
16 views

Reflection principle for walk possible steps right, left and stay

I need to use reflection principle for one dimensional walk with equaly possible steps right, left and stay. I would like to know if hold a similiar identity to that of question Is there an intuitive ...
-3
votes
1answer
39 views

what is the approximate probability that you win more than 120 times if you purchase 900 tickets? [on hold]

The fine print on an instant lottery ticket claims that one in nine tickets win a prize. What is the approximate probability that you win more than 120 times if you purchase 900 tickets?
0
votes
2answers
34 views

Show that $(\bar{X})^2$ is not an unbiased estimator for $\mu^2$

If $X_1, ... , X_n$ are $n$ identical distributed independent random variables each with mean $\mu$ and variance $1$. A little confused by this question. Is it asking for if $(\bar{X})^2$ != $\mu^2$....
0
votes
2answers
35 views

Random probability

So the story is my friend was playing Runescape and he was trying to get an item drop that had a ${1\over 128}$ drop rate so on average every $128$ monsters he slays one of the items will drop.he ...
0
votes
2answers
27 views

Bank Card ATM pins and probability

Bank ATM cards let you choose a four digit combination of integers from 1 to 9 given $9^4 = 6561$ different choices. If a person were to try 3000 choices or more generally $x \leq 6561$ for $x \in \...
-1
votes
1answer
23 views

Finding a constant in a joint probability density function

Given the following joint density function: \begin{equation} f (x,y) = \begin{cases} c(x+y)^2& \text{} 0 \le x \le 1, 0 \le y \le 1\\ 0 &\text{otherwise} \end{cases} \end{equation} I need ...
1
vote
2answers
23 views

Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
2
votes
1answer
70 views

Creating unusual probabilities with a single dice, using the minimal number of expected rolls

Problem I want to create an 'event' with probability of $\frac{1}{7}$ with a single dice as efficiently as possible (to roll the dice as little as possible). To give you some better understanding of ...
2
votes
1answer
36 views

determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
3
votes
1answer
17 views

probability question with marbles

I have $13$ different color marbles. One color is $5$ times likely to be chosen and another color is half as likely to be chosen. What is the probability that, $1$. you choose the marble that is 5 ...
0
votes
0answers
22 views

Conditional PDF for uniform distribution [on hold]

I have random variables X and Y. X is uniformly distributed over (0,12) and Y is uniformly distributed over (x/2, 3x/2). I want to calculate the conditional PDF of Y given X. The PDF of X is simply ...
1
vote
0answers
18 views

Multivariate normal distribution conditional probability question.

$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$We have that $X$ and $Y$ are random variables with a ...
0
votes
0answers
26 views

Proving that an integral of several cdf and pdf functions is increasing in a certain parameter.

Basic assumptions: $n\geq3$, $a\leq b\leq c$, $b$ is simply a dummy variable of integration, and $\rho\geq0$. $F(z)$ and $f(z)$ represent the usual general CDF and PDF (no specified distribution here)....
4
votes
1answer
50 views

conditional probability on zero probability events and conditional Radon-Nikodym derivatives

Consider a stochastic process $\{x_t\}_{t\in T}$ adapted to some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\in T},\mathbb{P})$ taking values in the state space $(\mathbb{R},\...
0
votes
1answer
31 views

How many different groups of 12 people can be chosen from a group of 30. More restrictions on details.

How many different groups of 12 people can be chosen from a group of 30. Note: the group of 30 contains: 2 people that will not work together (pick neither, or pick one, but not both) and 2 people ...
1
vote
1answer
50 views

Expectation of $|H - T|$

Using binomial approximation to normal distribution, find the expectation of $|H-T|$ where the $H,T$ are heads and tails of a fair coin and the number of tosses is large. Can anyone please tell me, ...
1
vote
0answers
48 views

probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
2
votes
1answer
36 views

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to?

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to- $\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}$ $\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$...
3
votes
2answers
56 views

How to precisely define a function that chooses randomly from a finite set?

Let $A = \{1, 2, \ldots, n\}$. I want to define a function that picks with uniform probability an element in $A$, so that $$f(A) = i \in A.$$ I don't know how to precisely define this mathematical ...
0
votes
3answers
29 views

Does $1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$ hold?

I am wondering does $1-\mathbb{P}(X_1>x_1, X_2>x_2)=\mathbb{P}(X_1\leq x_1,X_2>x_2)$? Even if $X_1$ and $X_2$ are dependent?
0
votes
1answer
16 views

Getting the joint function. What am i doing wrong?!?

we have that $f(x_1,x_2)=2(1-x_1)$ if $0≤x_1≤1$, $0≤x_2≤1$. And we have that $Y_1=x_1x_2$ and $y_2=x_1$ And i have to find the joint distribution of $y_1$, $y_2$:(f($y_1,$$y_2$)) and verify if this a ...
1
vote
1answer
20 views

conditional probability (discrete case).

I am not sure if I am doing this right.  We have this table $$\begin{array}{r:r|rr} & & X\\\hdashline & & 1 & 2\\ \hline & 0&.12 & .08\...
1
vote
1answer
54 views

Calculate the probability that the running total is exactly n. (homework help)

I am working through Harvard's public Stat 110 (probability) course. Question: A fair die is rolled repeatedly, and a running total is kept (which is, at each time, the total of all the rolls up ...
-3
votes
0answers
59 views

planning on trading, need mathematical edge [on hold]

I have been looking in to binary options trading, How It Works Retail trader (maybe me) goes to broker to trade binary options. If I trade that I think Euro/USD currency pair will go down, then I ...
1
vote
1answer
62 views

What is the probability of two-pair poker hand?

To start with, this question has never been asked as how I am going to ask: What is the probability that a five card poker hand will have two pairs (with no additional cards)? Example of two-...
1
vote
0answers
34 views

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
0
votes
0answers
15 views

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$ $o_i$ = observed $e_i$=expected (I removed $i$'s for ease) The solution is: $$G^2=2\sum o \log \frac{o}{e}$$ $$=2\...
0
votes
1answer
39 views

Coin toss related problem

What is the minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96? Is there any shortcut way to calculate this?
1
vote
0answers
19 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
0
votes
1answer
51 views

Show that $\frac{S_n}{n}\to 0$ in probability if $s<\frac{1}{2}$

Let $s\in\mathbb{R}$ and $X_1,X_2,\dots$ be independent random variables and with distributions: $$P(X_n=n^s)=P(X_n=-n^s)=\frac{1}{2}$$ Let $S_n=X_1+\dots+X_n$. Show that $$\frac{S_n}{n}\to 0 \text{ ...