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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Total Probability theorem and Bayes theorem

Two reinforced concrete buildings A and B are located in a seismic region. It is estimated that an impending earthquake in the region might be strong (S), moderate (M), or weak (W) with probabilities. ...
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0answers
10 views

Find all random variables whose distribution satisfies an equation

The problem I have to solve is formulated as follows: Find all random variables such that if $Y$ has the distribution $N(0,1)$ and $X, Y$ are independent then $X+Y$ has the same distribution as ...
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1answer
13 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
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1answer
20 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
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0answers
14 views

Estimate probablity: Chernoff bound

Two players $A$ and $B$ are playing following game: They throw cube. When thrown number $k$ and $k$ is even then player $A$ get $k$ points. When thrown number $k$ and $k$ is divisible by $3$ then ...
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1answer
26 views

What does it mean $\int_1^\infty\frac{F(y)}{y^2}\mathrm dy$?

Which type of functions will satisfy this? $$F: [1,\infty) \to [0,\infty)$$ $$\int_1^\infty \frac{F(y)}{y^2} dy \leq 1$$
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1answer
17 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...
2
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2answers
40 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
2
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1answer
13 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
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2answers
36 views

What is the logic behind the probability of getting 'four of a kind' in poker?

This hand ($5$ cards of $52$) has the pattern $AAAAB$ where $A$ and $B$ are from distinct kinds. The number of such hands is $\binom{13}{1} \binom{4}{4} \binom{12}{1} \binom{4}{1}$. The probability ...
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2answers
38 views

How can I reword this problem illustrating a scenario that needs Bayes Theorem to solve?

Taken from Stat Trek, an example explaining Bayes Theorm http://stattrek.com/probability/bayes-theorem.aspx Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent ...
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1answer
15 views

Help finding a probability density function

I am having a bit of trouble with this: Let $U=Z^2$ where Z is the probability density function of the standard normal distribution. So, $f_z(z) = \frac{1}{2\pi} e^{\frac{-z^2}{2}}$ I want to use ...
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0answers
15 views

Estimating probablity with using Chernoff inquality

I would like to estimate that probablity that $X_k>1250$. So, $X_k=Y_1 +...Y_{10^6}$ $Y_i = 1 $ with prob = $\frac{1}{10^3}$ $EX_k = 1000 $ Chernoff that I will use: $$P(X_k \ge (1+\epsilon)EX_k) ...
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0answers
29 views

Is this an easy conditional probability question?

Fifty-two percent of the students at a certain college are females. Five percent of the students are majoring in computer science. Two percent of the students are females majoring in computer science. ...
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1answer
24 views

Expected Value Coins Question

If I were to flip n coins and compute the product of the number of heads versus the number of tails what would be the expected value of this product? My logic: In n coin flips n/2 coins will be ...
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3answers
20 views

Chances of random number belong to a given set

I have 23 elements and 7 of them belong to a given set. 5 of these 23 elements will be picked randomly, I want to know the chances of at least one of those selected 5 elements belong to the ...
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1answer
23 views

Flipping an unfair coin n times

I’m flipping an unfair coin $n$ times. $\mathbb{P}[X=head]=p$ where $p \neq \frac{1}{2}$. What is the probability “head” appears an even number of times? Thank you in advance for your time an ...
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0answers
19 views

Repeated coin flips probability [on hold]

Assume in an experiment, one flips a coin $L$ times. This experiment is repeated T times. Assume the $k$'th flip for all possible $k$ values ($1 \le k \le L$) among all experiements. If the head ...
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3answers
39 views

Probability of drawing n distinct values out of {1,…,n^3}

I draw uniformly at random $n$ values out of $\{1,...,n^3\}$. I want to lowerbound the probability of getting $n$ distinct values.
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0answers
28 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
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1answer
13 views

Computing Average Number of Successes When Randomness is Involved

I am attempting to write a program that will compute the average amount of a particular product produced when randomness is involved. Let's say that I am trying to produce some widget. Whenever the ...
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0answers
18 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...
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0answers
10 views

How can I scale the covariance matrix which represent a gaussian distribution ? [on hold]

I have a model genrated by using GMM the output is the mean and covariance matrix .I need to scale the cov matrix .for example I want to double the elipse that represent this gaussian .
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1answer
27 views

Probability related question, Permutations, combinations [on hold]

Im doing a practice problem for an upcoming test, I had a hard time figuring out this question, could anyone walk me through it?
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0answers
86 views

Is there a real problem to which $1$ radian is the answer?

I can't recall if I've ever seen any problem related to angles, in math or engineering books, that would result in an answer like $$\alpha=1 \ \ \text{radian}.$$ The answers to such questions, I ...
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1answer
39 views

What does it mean that an expected value does not exist?

$X$ is a random variable with pdf $f$ and $g: \mathbb R \to \mathbb R$ is a measurable function. Before I start operating with $E[g(X)]$ I need to show that it exists. What does it take to show it? ...
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0answers
11 views

Deducing results about continuous time random walks from the corresponding discrete time result

Is there any standard way to prove results about continuous time random walks from the corresponding results for discrete time random walks? Specifically, my problem is that I was reading Lawler and ...
0
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1answer
11 views

What is the probability that $x$ will not work due to failure rate $0.0111$

I've tried using the probability mass function for binomial distribution in this case but it seems to not be the appropriate approach unless I calculated wrong. How am I supposed to approach this ...
2
votes
1answer
49 views

Expected number of red balls removed from an urn before the first black ball

Question: An urn contains n+m balls of which n are red and m are black. They are withdrawn from the urn one at a time and without replacement. Let $X$ be the number of red balls removed before the ...
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0answers
23 views

Cube of Brownian motion [on hold]

Find all $H_t$ so that: $B^3_T = \int_0 ^T H_t dB_t$ $\int_0^T B^3_tdB_t = \int_0^T H_t dB_t$ where $B_t$ is a Brownian motion.
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1answer
40 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
3
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1answer
20 views

Using Conditional Jensen inequality proof the following

$X_1,X_2,\ldots,X_n$ are i.i.d. random variables, $X_1>0$, $E[X_1]=\mu$, $E[X_1^k]<\infty$ for $1<k \leq2$. Proof: $$ E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq ...
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0answers
27 views

conditional probability proof 3 varables [on hold]

Suppose that $\mathcal a$ ,$\mathcal b$ and $\mathcal c$ are dependent variables. $$\mathbb P(a \mid b) = \sum \mathbb P(a \mid b,c) \ \mathbb P(c \mid b)$$ can anyone explain it how we get it?
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2answers
23 views

Rolling dice probability by solving inequlity

I was trying to solve a problem where I have to find the probability of the sum of $\mathcal 3$ rolls of a die being less than or equal to $\mathcal 9$. In order to solve the problem I try first to ...
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0answers
21 views

Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
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0answers
21 views

How to use joint probability density to check for independent events?

Suppose that the joint PDF of $X$ and $Y$ is as follows: $$ f(x) = \begin{cases} 24xy & \text {$x \geq 0, y \geq 0, x+y \leq 1$}\\ 0 & \text {otherwise ...
1
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1answer
17 views

Covariance of $2$ variables

I am given two random variables $X$ and $Y$. I am also given that $\mathbb{E}(Y|X)=\mathbb{E}(Y)=\mu_y$ and $\mathbb{E}(X)=\mu_x$. So if I need to calculate the covariance of $X$ and $Y$, ...
5
votes
1answer
34 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
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1answer
23 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
0
votes
2answers
15 views

Multinomial Coefficients Dice Problem

If 7 balanced dice, are rolled, what is the probability that each of the 6 different numbers will appear at least once? My attempt: $p=\frac{7!}{2!6^6}$ So if 6 different numbers need to appear, ...
2
votes
3answers
34 views

Probability of balls in boxes

If $12$ balls are thrown at random into $20$ boxes, what is the probability that no box will receive more than $1$ ball? So my book says the answer is: $\displaystyle \frac{20!}{8!20^{12}}$ However ...
2
votes
1answer
38 views

Calculate the mean, the median and the quartiles.

Let $D=\{(x,y):x>0,x^2+y^2<1\}$ and let $(X,Y)$ be the random variable with the density: $$f(x,y)=\frac{2}{\pi}1_{D}(x,y).$$ Let $Z=\frac{Y}{X}$. Calculate the mean, the median and the first and ...
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1answer
27 views

Probability and Statistics (Normal Distribution)

Having trouble with the last part of this question. Not sure how the man would divide his pile of vouchers? It seems that you could interpret this question in a lot of ways. Any tips would be ...
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0answers
10 views

Obtaining the Log-logistic distribution from a truncated logistic distribution

Let $$f(x) = \frac{e^x}{(1+e^x)^2}~,~ -\infty \lt x \lt \infty~~~~~(1)$$ be the standard logistic pdf of a random variable $X$. Then one can obtain the pdf of the log-logistic distribution via the ...
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0answers
20 views

Martingale Ideas in Elementary Probability [on hold]

There is a non-symmetric version of the probability model in which the probability of success on each trial is "$p$" and the probability of failure on each trial is "$q$" and $p+q=1$. The probability ...
0
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1answer
29 views

Calculate the probability given by three random variables

Let $X_1,X_2,X_3$ be IID random variables, each with the density $$f(x)=x e^{-x}\cdot 1_{(0,\infty)}(x).$$ Calculate $P(X_1+X_2+X_3>4,X_1+X_2<4)$.
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0answers
11 views

Question on Standard Brownian Motion [on hold]

What is the following probability $P [ W(2) >0 \ \text{and}\ \ W(1) <0]$?
2
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2answers
27 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
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0answers
17 views

Show that for any geometric random variable $X$ and parameter $p, \mathrm{Pr}(X < t) = 1 − p^t$. [on hold]

How to prove the above stated equation? I tried the following : Pr⁡(X(i=1)^(t-1)▒〖Pr⁡(X=i)〗 =∑(i=1)^(t-1)▒〖p(1-p)i-1〗 =1-(1-p)t-1
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0answers
31 views

Conditional Probability Proof for three events

For any 3 events $X, Y$ and $Z$ where $\Pr Z) > 0$, it is required to prove that $$\Pr ((X \cup Y) \mid Z ) = \Pr(X\mid Z) +\Pr(Y\mid Z) - \Pr ((X \cap Y) \mid Z)$$ I am not able to prove ...