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
47 views

Finding the general pattern form

Suppose a Markov chain given as follows. $p_{ii}=1-3a$ And $p_{ij}=a\ \forall i\ne j$ where $P=(p_{ij}), 1\le i,j\le 4$. Find $P_{1,1}^n$. Attempts: I have tried to compute for the case $n=1, 2, 3$. ...
4
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1answer
89 views

Expected number of times before winning the lottery $n$ times

Let $p,n$ be positive integers. Suppose that every time you buy the lottery, you have a $\dfrac1p$ chance of winning it (independently of other times). What is the expected number of times you have to ...
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1answer
81 views

Counting/ Probability

Please please please... explain this to me. assume that a die is tossed twice and the numbers showing on the top faces are recorded in sequence Determine the elements in each of the given events a- ...
1
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1answer
205 views

Probability that the majority of 3 classifiers are wrong

3 classifiers. Each classifier has $0.7$ accuracy and makes its error independently. How do I calculate the probability that the majority of three classifiers are wrong? how I'm trying to solve it: ...
2
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1answer
33 views

Deriving the equation for the probability of a prime number

A couple of months ago I stumbled upon an equation regarding the probability of having a prime number adjacent to "x". If I remember correctly it was: $\frac {x}{ln x}$ ,or something along those ...
2
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1answer
133 views

Help understanding a counting and probability exercise

I need help in trying to understand the answer to this exercise. [Question] A club is considering changing its bylaws. In an initial straw vote on the issue, 24 of the 40 members of the club favored ...
3
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1answer
77 views

Optimal consumption policy

I start with an initial capital C and at the beginning of day $n=1,...,N$ I observe the random variable $X_n$, where $\mathbb E X_n=\mu_n$. The $X_n$ are independent. I also choose $c_n$ on day $n$, ...
3
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2answers
59 views

Are A and B conditionally independent

Are A and B conditionally independent given the class label? I calculated that $$P(A=1) = \frac{1}{2}$$ $$P(B=1) = \frac{2}{5}$$ $$P(A=1,B=1)=\frac{1}{5}$$ My answer is yes. I do it by anding ...
0
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1answer
83 views

Random Process not mean square continuous

Does a WSS (Wide Sense Stationary) process exist which is not mean square continuous? If so, can you give me an example. Note: A WSS process is mean square continuous iff the autocorrelation ...
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2answers
89 views

Probability of winning prizes

Special Drawing where you pay \$10 to play. There were 500 tickets sold a possible 5 prizes won. The first place prize gets \$500. Four second place prizes award \$100 each. What is a prob. ...
2
votes
1answer
77 views

Optimal Investment Strategy

I am not sure to solve the following investment problem: I have an investor which receives an income $I_n\ge 0$ at the start of year $n$. The investor chooses a proportion $p_n\in[0,1]$ of this in ...
0
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1answer
68 views

0-1 Law: Applications?

The question is more open to a debate rather than a mathematical explanation: If $(A_n)_{n\in \mathbb{N}}$ is a sequence of $\sigma$-algebras. $\mathfrak{A}_n := \sigma(\bigcup_{m\ge n} A_m)$ is the ...
2
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1answer
57 views

Probability: Why assume an equal relevant magnitude of two mutually exclusive events for P(A|B)?

While reading Grinstead and Snell’s Introduction to Probability p.134, I came across the following: "Let Ω = {ω1,ω2,...,ωr} be the original sample space with distribution function m(ωj) assigned. ...
3
votes
1answer
108 views

Unbiased estimators for the moments of 2 not independent random variables

Let $X$ and $Y$ be two non independent random variables. Suppose to generate $n$ realizations of both variables, and indicate with $(X_i, Y_i)$ their values. Also, let's pose that $X$ and $Y$ are non ...
2
votes
1answer
111 views

My Odometer, Speedometer, and the Time

I was driving from home to university earlier this week when I glanced at my dashboard just soon enough to notice the odometer tick up while simultaneously registering my speed and the time. I ...
4
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1answer
94 views

Conditional expectation by $\sigma (G_n,Y)$ when $Y$ is $G_\infty$-measurable

Let $G_n$ be a filtration (an increasing sequence of sigma-algebras), $Y$ a random variable that is $G_\infty$-measurable, and $X$ a random variable. Is it true that in $L^2$-norm, $$ \mathbb{E}[X ...
2
votes
1answer
57 views

Probability with 13 trees

A company is planting trees and we know that 90% of the trees survive. What is the probability that from 13 trees: 1, at most $10$ survive 2, at least $10$ survive 3, exactly $10$ trees survive.
2
votes
2answers
58 views

Throwing of non-fair dices

Given that Mark has a non-fair dice with probability of getting a $6$ is $p$, and the probability that Alex getting a $6$ is $2p, (0 < p \leq 0.5)$, what is the probability of Mark winning the ...
2
votes
1answer
833 views

Find unknown value in probability density function

Suppose that a random variable $Y$ has a p.d.f. given by $$f (y) = ky^3e^{-y/2}$$ when $y > 0$, and otherwise 0. Find the value of $k$ that makes $f(y)$ a density function. I found that ...
1
vote
1answer
22 views

Are these disjoint/dependent?

Given $P(A) = 0.7, P(B) = 0.6, P(A^c | B^c) = 0.25$, are: I) $A$ and $B$ disjoint? II) $A$ and $B$ dependent? So, what I said: $I)$ Since $P(A) + P(B) = 1.3 > 1$ then $P(A \cap ...
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2answers
590 views

Expected value and variance on exponential distribution

The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C = 100 + 40Y + 3Y^2 relate the cost C of ...
5
votes
1answer
434 views

Proof on Brownian Bridge

PROBLEM Let $U_t$ be a Brownian bridge on $[0,1]$ and let $Z$ be a standard normal random variable independent of $U_t$. $(a)$ Prove that the process $W_t = U_t + tZ$ is a brownian motion. $(b)$ ...
0
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2answers
88 views

Conditional Probability coin tossing

I was going through some exercises on probability and came across a question. Two people A and B are tossing a fair coin with A tossing first. The process is repeated till someone gets a heads. What ...
5
votes
1answer
792 views

Expected value of infinite sum

$x_1, x_2, \dots, x_n, \dots$ - independent random variables. Is it true that $$ \sum_{i = 1}^{\infty}Ex_i = E(\sum_{i = 1}^{\infty} x_i) $$ ?
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1answer
21 views

Need someone to quickly confirm whether I have this expected value correct

I am given that $E(X)=2$, $E(Y)=4$, $E(X^2)=6$, $E(Y^2)=20$ and $E(XY)=1$. I am asked to calculate $E((3X-2)^2)$ - would I be correct in representing this as $9E(X^2)-12E(X)+4$ so my answer would be ...
0
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1answer
170 views

Probability Brownian Motion - dependence

Does anyone know how to calculate $P(Z(3)>Z(2), Z(2)>0)$ if $Z(3)$ and $Z(2)$ are on the same sample path, i.e. not independent? I found a solution for the case $P(Z(2)<0, Z(1)<0)$ in ...
0
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2answers
128 views

Expectation of absolute value of stationary time series

Let $Y_t$ be a stochastic process (time series). We consider stationarity as follows: $Y_t$ is said to stationary if the mean $\mu_t = \mathbb{E}(Y_t)$ is constant (given $\mathbb{E}|Y_t|<\infty$) ...
1
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1answer
133 views

covariance after conditional expectation (projection)

Suppose two random variables $X$ and $Y$, and their covariance could be $E(XY)$ if we simply assume their expectations are zero. Now, we take the conditional expectation of both: $\xi = E(X\mid Z)$ ...
0
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1answer
97 views

Rain drops falling into a cup via a Poisson process and leaving by exponential decay

Imagine I place a cup out in the rain, and that point-like rain drops arrive in the cup via a Poisson process with rate parameter $\lambda_1$ (the time for a single cloud particle to release a drop of ...
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2answers
89 views

Finding the density function $f_X$ from the distribution function

Can someone help me find the density function $f_X$ for $X$ and hence find $E(X)$ and $Var(X)$ of the following distribution function $F_X$ given by: $F_X(x)=\begin{cases} 1-(1+x)e^{-x} & x>0 ...
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1answer
118 views

counting and probability question - help needed

I am stuck on how to start this exercise. Any help is welcome. An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. Suppose the exam instructions specify ...
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1answer
1k views

Conditional distribution of binomial random variables is hypergeometric

Let's say $X$ and $Y$ are binomial random variables with parameters $n$ and $p$ and $X+Y=m$. I want to show that the conditional distribution of $X$ if $X+Y=m$ is a hypergeometric distribution. I'm ...
2
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2answers
2k views

Finding joint density, marginal density, conditional density of bivariate normal distribution

Suppose that (X,Y) has a bivariate normal distribution. You know that Y is a standard normal random variable and that the conditional distribution of X given that Y=y has mean 3y-4 and variance 7. ...
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1answer
171 views

Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what ...
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2answers
150 views

Question How many beads of each color will each necklace have…

We have 72 red beads,81 yellow beads and 99 white ones and we want to make a necklace,and each necklace should have 9 beads.How many beads of each color will each necklace have??
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0answers
74 views

Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$. Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I ...
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1answer
34 views

Joint distribution proof

I am trying to study for an exam and I am kind of lost on how my professor came to a particular result on his practice exam. Let $W$ be an exponentially distributed random variable with $\lambda = 2$ ...
2
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1answer
369 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 ...
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1answer
173 views

Good Probability Practice Problems

I'm looking for a good probability textbook with lots of worked out examples and problems to prepare for my course's final exam. I'm in an introductory probability class in college, and we've covered ...
2
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1answer
54 views

Number of ways to rearrange a line of $n$ marbles

My friend challenged me to solve the following problem, and after having thought about it for a long time and not being able to find the answer, I decided to give up. His explanation which followed ...
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4answers
44 views

Probability of getting something with a low probablity

If there are 100 marbles in a bag (1 red one, 99 green ones), then the probability of picking the red one is 1/100. But if I do 100 trials then I believe it is likely that I will pick the red one at ...
0
votes
1answer
26 views

Continuous distribution with probability density function $f_X$ question

Let $X$ have a continuous distribution with pdf $f_X$ given by: $f_X(x)=\begin{cases} 0.5+x & 0<x<1 \\ 0 & otherwise. \end{cases}$ I must find $E(X)$ and $Var(X)$ and also find the ...
0
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1answer
115 views

Correlation coefficient question regarding coin tosses

So a coin is tossed 10 times in this question. $X=$ number of heads in the first 5 tosses $Y=$ number of heads in total $T=$ number of tails in the first 5 tosses I am asked to calculate the ...
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0answers
130 views

Analogy of $[0,1]^n$

A probability maps a set (such as 1 dimensional path) to $[0,1]$. Is there a sufficient and necessary analogy of $[0,1]^n$ to a $n-$dimensional cube ? or to $n-$dimensional other well known objects ? ...
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1answer
74 views

How many iterations does it take to cover a range with random values?

Let's say I have a random number generator that generates integers uniformly from 0 to n-1 (where n is some positive integer). What is the expected number of iterations after which all the values ...
1
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1answer
38 views

Constant Sample Variance

If the sample variance $S^2$ is constant, can we say the population is constant? If so, how to prove this?
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1answer
1k views

Solution to probability problems

could you please help me? I know the correct solutions which are different from mine, and I absolutely agree that they are correct. However, I cannot find out what's wrong with my solutions that ...
0
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1answer
41 views

the probability of rolling a die

There is a die with six faces numbered consecutively from 1 to 6. What is odd about it, is that the probability of rolling the face with number k on it is c*(q^k), where c is a constant, and q = 0.9. ...
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0answers
124 views

What method to use for this(poison binomial or binomial distribution?

The experiment E is to take a required computer science course. The outcomes are the grades (A, B+,B, C+, C, D, F). Suppose the experiment is repeated 5 times, once for each of CS111, CS112,CS113, ...
3
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1answer
46 views

$\frac{1}{n}\sum_{i=1}^nZ_i \rightarrow \int_0^1f(x)dx$

PROBLEM Let $X_1,Y_1,X_2,Y_2,...$ be a sequence of independent random variables, all of which distributed uniformly on $[0,1]$. Let $f: [0,1] \rightarrow [0,1]$ be a continuous function. Define ...