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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1answer
17 views

Expectation of a function of weighted sum

Set of weights $w_j$, set of i.i.d. random variables $X_j$ and $f(y)$ is a decreasing function in $y$. I want to claim that if $\sum w_i<\sum w^{\prime}_i$ Then $\mathbb E f(\sum ...
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1answer
4 views

How to find Reliability of a rectangular distribution function?

Assume that the failing of a device is equally probable within an interval [a,b] such that the fault density is: f(x) = {1/b-a if a<= t <= b ...
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1answer
24 views

Calculate the Probability in competition

The committee RAM competition knows from experience that the probability of successfully Contest is 0.95 for the student who has grade "very good" in BAC test , 0.5 one who has 'Good' in ...
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2answers
29 views

Non-Probabilistic Argument for Divergence of the Simple Random Walk

The simple random walk is one starting at $0$ with steps of $-1$ and $1$ with equal probability. Is there a proof not involving (too much) probability - preferably number-theoretic - of why this walk ...
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3answers
17 views

How many ways to fill six broadcast slots for 3 adertisements which are to be shown twice

A television director is scheduling a certain sponsor’s commercials for an upcoming broadcast. There are six slots available for commercials. In how many ways may the director schedule the commercials ...
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1answer
17 views

What is the probability of choosing 5 random students whose (individual) grade is higher than 1149 points?

I know this isn't that hard, but I have been looking and I don't know how to solve it. The number of students whose grade is higher than 1149 is 44, and the total of students is 135. If the question ...
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3answers
73 views

The probability that after repeated random drawing from an urn, all balls left in the urn will be red

Problem An urn contains $p$ red and $q$ green balls. Balls are drawn one by one till balls left in the urn are all red. Prove that the probability of this event is $\dfrac {p}{p+q}$. Please note that ...
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2answers
27 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
3
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3answers
32 views

Probability of having 4 aces after taking turns to pick cards

I've started to learn probability and I get stuck with the following problem: My friend and I are playing a card game with 36 unique cards. There are four suits (diamonds, heart, clubs and spades), ...
2
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1answer
20 views

Probability question with Geometric random variable

Sir Lancelot and Sir Galahad are doing a shoot out, in which they try to shoot each other while shooting in the same time at each other. The probability of Sir Lancelot to hit Sir Galahad is 0.5 and ...
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1answer
26 views
0
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1answer
28 views

After $n$ sticks are broken into two parts each, they are joined again randomly. Find the probability of them being joined in a certain way

Each of $n$ sticks are broken into a longer and a shorter part. Out of these $2n$ parts, $n$ sticks are formed again by joining any 2 parts randomly. Find the probability that a) The parts will be ...
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3answers
57 views

Birthday paradox, huge numbers

Pick x random "birthdays", say $10^9$. What are the chance of a collision, given $2^{160}$ possible "days"? I'm trying to estimate the collision rate of sha1 hashes, but the calculation is too big ...
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1answer
25 views

Probabilities of calling with Cell phones

Cell phones perform transfers as they move from cell to cell. During a call, a phone can make zero transfers ($H0$), a transfer ($H1$) or more of a transfer ($H2$). Additionally, each call is ¨long¨ ...
4
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1answer
67 views

Probability of drawing a run of a specific color from an urn with two colors of balls

I was sent a puzzle involving an urn with 128 white balls and 288 black. If the balls are drawn without replacement until the urn is exhausted, what is the probability that a sequence of 10 or more ...
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2answers
24 views

Schwartz Inequality (probability) - first step in proof

I'm trying to understand the Schwartz Inequality for random variables, which states $$(E[XY])^2 \leq E[X^2]E[Y^2] $$ The solution states that we can assume $E[Y^2] \neq 0$ because if this were the ...
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2answers
92 views

Basketball shots and stopping rule

You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can stop whenever you see fit and ...
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0answers
24 views

Statistics: How would I correlate many variables to a few coefficients?

I'm trying to predict the "strength" vs "tempertature" and "time" curves of some chemical compounds as a function of the concentrations of their component substances. I have 20 different substances. ...
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2answers
55 views

Weird probability question, probability on successive trials [on hold]

In doing practice problems for an upcoming exam, I came across this question, which frankly has me stumped. In a laboratory experiment an attempt is made to train an animal to turn right in a ...
6
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1answer
115 views

Sum of average reciprocal of which random variable converges to a Cauchy distribution?

If $(X_n)_{n\in\mathbb{N}}$ are independent identically distributed random variables with density $f$ even, continuous in $0$ and such that $f(0)>0$, then $$\frac{1}{n}\left(\frac{1}{X_1}+\dots + ...
1
vote
1answer
31 views

Odds/Probability

I don't know the proper terminology but I would like to get a primer on odds and probability. If the odds are set at 1 in 500 what is the probability of winning on the first try (0.02%?), after 500 ...
0
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1answer
35 views

A simple problem on probability

Suppose we have a train running on a railroad from $A$ to $B$. The railroad is N Km long from the point $A$ to the point $B$ and the speed of the train is $v$ $Km/h$. We have two situations: in the ...
0
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1answer
14 views

infer the initial state from draws

I went through binomial distribution and Chi-square test etc and got confused further. This question might be very basic and simple. I have three states (Combination of two colors, both has equal ...
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5answers
53 views

3 distinct numbers are selected from $\{1,\ldots,9\}$. What is the probability that 9 is selected?

3 distinct digits are randomly selected from the set of nine digits [1-9]. What is the probability that 9 is selected. I thought that the probability should be (1/9) + (1/8) + (1/7) since you have to ...
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0answers
16 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
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0answers
38 views

Probability of getting head in coin flip [duplicate]

Suppose a football match is going to be started. The referee should flip a coin to give the ball to one of the football teams. In front of the two captains the referee flips the coin two times and ...
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0answers
28 views

Probability of guessing Date of Birth

Is it possible to give the probability of guessing the Date of Birth of a complete stranger i.e. to the exact year, day and month? If so, what would that be?
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votes
1answer
36 views

Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ converges to 1 in some neighbourhood of $t=0$.

$X_n$ is a sequence of random variables, and $\{\varphi_n(t)\}$ is the corresponding sequence of characteristic functions. Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ ...
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0answers
19 views

Approximating the probability of an event by finite-dimensional distributions

Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following ...
1
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1answer
268 views

Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
1
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2answers
40 views

Conditional Expectation Discrete and Continuous

Find $E[X]$ and $Var[X]$ So for the expectation so far I got that: $$E[X] = E[X|N=n]P(N=n) = \large\frac{n+1}{\lambda} \frac{\lambda^{n}}{n!}e^{-\lambda}$$ but for conditioning on both a discrete ...
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0answers
15 views

On the geometry description of the GSR riffle shuffle model

In 1992 Diaconis and Bayer announced their famous result which is now a well-known folklore: Seven shuffling is enough to randomize a deck of cards. One of the key ingredients in their proof is that ...
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2answers
34 views

Multivariate distribution

Let $X_1, X_2, X_3$ be independent random variables with normal distribution $n(0, \sigma^2)$. Let $Y = (X_1^2 + X_2^2 + X_3^2)^{1/2}$. Find the density of $Y$. I tried finding the densities of ...
1
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1answer
28 views

E[X] and E[X^2] with Conditional Expectation

$\newcommand{\E}{\operatorname{\mathbb E}}$ $\newcommand{\Var}{\operatorname{\mathbb Var}}$ If $\E[X] = {^1\!/\!_3}(\E[X\mid Y=1] + \E[X\mid Y=2] + \E[X\mid Y=3]) = 10$ Where $\E[X|Y=1] = 2,\; ...
2
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1answer
63 views
+50

Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
0
votes
1answer
51 views

$X$ ~ $xe^{-x}$ for $I (x > 0)$. $Y$, given $X = x$, is uniformly distributed over $(0,x)$. Find…

a) $f_Y (y | X = x) =$ ? $\,\,\,\,\,\,\,E (y | X = x) =$ ? b) $f(x,y) =$ ? c) $f_Y (y) =$ ? d) $f_X (x | Y = y) =$ ? e) $E(X | Y = y) =$ ? My work: a) $f_Y(y|x) = \frac{1}{(x-0)} = ...
0
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3answers
48 views

Should I pick the higher dice?

Assuming I start with $n$ dice that have been rolled once, is it beneficial to choose the higher dice when I roll less than $n$ dice again (assuming I want a high roll)? In some board games, dice ...
-1
votes
1answer
97 views

Find the warranty period such that the battery is replaced under warranty 0.5% of the time

Problem The mean life of a Chevy Volt battery (normally distributed) is $1000$ hours and the standard deviation is $100$. How many hours should GM warranty the battery for so that it has to replace ...
-1
votes
1answer
186 views

Expected utility and St. Petersburg paradox

Can someone explain to me how they get the $10.94$ at the Expected utility theory section of the solutions to the St. Petersburg paradox? My problem is that they use a formula to calculate the ...
0
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1answer
16 views

Probability inequality exchanging sum with cardinality

Let $P_{XY}$ be the joint probability distribution of discrete random variables $X$, $Y$. Then I would like to prove the following inequality: $$ \sum_{y}\max_xP_{XY}(x,y)\leq |Y|\max_xP_X(x) $$ ...
2
votes
2answers
21 views

Determining the maximum % below average

Is there a way to determine the maximum percentage of values that fall below the average in a given sample? How would someone go about this? How does this relate to what Markov's inequality and ...
1
vote
3answers
44 views

Can this conditional probability be answered using Bayesian Theorem (or at all) with the information given

I have a conditional probability problem I'm unsure can be answered given the information I have - as such I'm unsure if Bayesian Theorem is the way to answer it, or if the answer is staring at me in ...
1
vote
1answer
212 views

Distribution function (CDF) of the sum of two random variables + law of iterated expectations

I'm taking my first probability class, and we're studying sums of independent random variables. We're using Ross's First Course in Probability. It states the definition of a convolution, but doesn't ...
4
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4answers
211 views

What does “twice as likely” mean?

Once in a while I hear people say something like X is twice as likely as Y. What they usually mean is: $$p(X) = 2 \cdot p(Y)$$ and - in the context they refer to - they usually have $p(Y) < ...
0
votes
1answer
13 views

Combining Random Variables to get the Variance

So $$\operatorname{Var}(aX + bY) = a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y).$$ But I have also seen: $$\operatorname{Var}(aX - bY) = a \operatorname{Var}(X) + b \operatorname{Var}(Y).$$ ...
0
votes
1answer
75 views

False Acceptance Rate [on hold]

Consider you have a fingerprint database containing the fingerprints of every person living in a country. To simplify the calculations we will assume there are $5,500,000$ people. Suppose the false ...
0
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0answers
14 views

Is there a bound on largest eigenvalue for covariance matrix of discrete random variable?

I have a random variable $Z=(Z_1,\ldots,Z_p)$. Each component can take values in {-1,0,1}. Is there a way to bound the largest eigenvalue of Cov(Z)? Actually, I have a latent multinormal variable ...
4
votes
2answers
128 views

Is it possible to be “too good” at Spider Solitaire?

There was a similar question here: Losing at Spider Solitaire However, what I'm asking is different. The game has a rule that it would not deal the next ten cards, unless there is already a card in ...
4
votes
2answers
519 views

Outcome of rolling a fair die 6 times

I'm failing to understand how to come to the answer to this question. If you roll a fair die six times, what is the probability that the numbers recorded are $1$, $2$, $3$, $4$, $5$, and $6$ in any ...
4
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2answers
72 views

Probability of Sum of Independent Events Exceeding a Value

Suppose I have $n$ random number generators. Once an hour, on the hour, each one generates a random real number $x_k$ such that $0 \le x_k \lt \infty$. Each generator produces its values according to ...