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
49 views

Conditional probability plane problem

I was presented with this problem and am not sure where to take it. A plane is missing and is presume to have equal probability of going down in any of 3 regions. If a plane is actually down in ...
1
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1answer
28 views

Ratio between normal distributed and gamma distributed variables

Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed?
1
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1answer
28 views

Compute Distribution of Linear Function

Let $Y = aX + b + \varepsilon $ , $X \sim \exp (\theta )$ , $\varepsilon \sim N(0,{\sigma ^2})$ , $X$ and $\varepsilon $ are independent How we can find distribution of $Y$? Thank you.
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0answers
9 views

average dirichlet distribution [on hold]

Is it possible to combining 2 Dirichlet distribution averaging their values? The resulting probability distribution is still a Dirichlet distribution? If not how can I merge 2 similar Dirichlet ...
0
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1answer
20 views

How to compute $P(X\leq Y)$ and $E(X^2 Y)$ with given probabilities [on hold]

Given $P(X=1, Y=0) = 0.1; P(X=1, Y=1) = 0.1; P(X=1, Y=2) = 0$ and $P(X=2, Y=0) = 0.2; P(X=2, Y=1) = 0.4; P(X=2, Y=2) = 0.2$ How do I compute $P(X \leq Y)$ and $E(X^2 \cdot Y)$?
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0answers
35 views

how can i get this probability expression? [on hold]

A deck of $n=10$ cards is numbered from $1$ to $10$. The cards are shuffled and laid down from left to right, face up. Order each of the five successive pairs of cards. Each of these five pairs ...
2
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1answer
31 views

How to prove this expectation equality?

How to prove this expectation equality? I am studying probability theory by myself and I find it hard. Thanks!
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0answers
18 views

A binomial-ish probability calculation

The probability $p_n$ that $n$ customers visit a supermarket in one day is $p_n=p^nq, n\ge 0$ where $p+q=1$. Also, on an average two out of three customers buy a certain type of item. The probability ...
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0answers
24 views

“Taking expectation” to yield conditional probability

This argument is taken from Resnicks Adventures in stochastic processes and let $T _{\infty } < \infty $ denote that an infinite number of transitions in a continuous time markov chain has occurd ...
0
votes
2answers
47 views

Elementary problem in Probability

A disease is spreading across the city of Rome. If the vaccine is taken, there is $ 95/100 $ probability that you won't be infected. Independently, there is $ 25/100 $ probability that you won't be ...
0
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1answer
30 views

Sum of $X_1,\ldots,X_n \sim \mathrm{exp}(1)$ i.i.d. random variables

Let be $X_1,\ldots,X_n\sim\mathrm{exp}(1)$ i.i.d. and $T_n =X_1 +X_2+\cdots+X_{n}$ Show with induction that the density of $T_n$ is $ \frac{1}{(n-1)!}a^{n-1}e^{-a}\,da $
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1answer
36 views

Even product on 5 dice rolls

A fair die (d6) is thrown five times. What is the probability that the product of the five scores is even? I have tried the following approaches. The first seems unnecessarily complex and also wrong: ...
0
votes
1answer
15 views

Combinatorics Forming a word from set of Alphaphets

A sequence of 15 random draws, one at a time with replacement, is made from the set { A,B,C ...,X,Y,Z } of the English Alphabet(26 Alphabets in total). What is the probability that the string ...
0
votes
1answer
31 views

Relationship between a distribution function and the truncated distribution function

Let $F(x)$ be a distribution function and $G(x)$ be $F(x)$ truncated on some interval $(a,b)$. I want to show that: $$G(x)=\frac{F(x)-F(a)}{F(b)-F(a)}, a<x \leq b$$ I want to do this by using ...
0
votes
1answer
40 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
1
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2answers
34 views

How to prove the above expectation inequality?

If $\mathbb{E}[|X|^k]<\infty$ then for $0<j<k$, $\mathbb{E}[|X|^j]<\infty$, and furthermore $\mathbb{E}[|X|^j]\leq(\mathbb{E}[|X|^k])^{j/k}.$ How to prove the above expectation ...
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1answer
36 views

some question about expected value

Let $X$ be a non negative random variable. We Know that $E(X)=0$. Is that correct that $X=0$ for some $X$. And more general: Is there a point in the probability space for which E[X]≤X and a ...
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4answers
48 views

Question from probability theory

I have this question: A person has three children with at least one boy. Find the probability of having at least two boys among the children. EDIT* --> My intuition about the problem is this-- the ...
0
votes
2answers
110 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
1
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1answer
16 views

Proving the Probability of a Subset

I am trying to learn probability and came across the following question. The conclusion proposed seems obvious but I am unsure on how to approach it. Any pointers in the right direction would be ...
2
votes
1answer
26 views

Probability when spinning a disk

A player spins a disk with three regions. The first region has the number 3. The second region has the number 5. The 3rd region has the number 4. On every spin, each of the numbered sections has a ...
2
votes
2answers
46 views

Multiple Weighted Coin Toss Probability

1) Is it possible to weight two coins in such a way that if the two coins are tossed, the three possible outcomes (both heads, both tails, or one head and one tail) all have probability $\frac{1}{3}$? ...
0
votes
2answers
38 views

Find expected value of F(N)

If we are given that a variable X is defined as X=rand() % N Here rand() returns an integer between 0 and $10^{100}$ (inclusive) uniformly at random. Now we ...
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0answers
39 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
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0answers
55 views

Ito integral solution

I need some help on how to compute the stochastic integral \begin{align} \int_{0}^{t}\frac{1}{\alpha-u}dW(u) \end{align} where $\alpha>0$.
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5answers
59 views

Counting principles question - team photo

A rugby team consists of 8 forwards, 7 backs and 5 subs. They all line up at random in one row for a picture. What is the probability that: a. the forwards are all next to each other? b. no two ...
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0answers
25 views

function of a random variable problem 2 [on hold]

Let $U$ be a continuous random variable with uniform distribution over $[0,1]$. Define $X$ by $$X=\operatorname{Int}\left(\frac{\ln(1-U)}{\ln(1-p)}\right)+1$$ where $\operatorname{Int}(x)$ is the ...
-2
votes
0answers
23 views

Probability of no person being present on a website [on hold]

Given that an average of 2 people are present on a website X at any given minute. What is the probability that no person is present on the website X in a 5 minute interval window (they need to take ...
0
votes
2answers
25 views

Consider the word 'PARTING'. What is the probability that a 4 letter sequence from this word contain the letter 'P'?

So far, I only have $_7C_4$ as the total amount of ways to arrange this (35). The answer is 4/7, and I can't really see how you get there.
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2answers
19 views

Probability - at least question

As part of a promotion a toy is put in each packet of crisps sold. There are eight different toys available. Each toy is equally likely to be found in any packet of crisps. David buys four packets of ...
0
votes
1answer
30 views

Ordered sequences of integer with fixed sum

Let $I_S = \{0, 1, \ldots, S\}$, with $S \geq 1$. Consider all the ordered sequences of length $L \geq 2$ in $I_S^L$ such that the sum of all the terms is equal to $S$. Let $N(L,S)$ be the number of ...
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1answer
29 views

Probability - pmf for alternating probabilities

Two people, Amanda and Bella, will have consecutive arm wrestling matches until one of them wins two matches in a row and is declared the winner. Amanda wins a given match with probability ...
0
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2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
-1
votes
1answer
24 views

function of a random variable problem [on hold]

Let $X$ be a random variable uniformly distributed over $[a,b]$. Let $Y=(X-c)^2$ where $c$ is the constant such that $c \epsilon (a,(a+b)/2)$. Find the density of Y. Please help me.
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0answers
22 views

Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
0
votes
1answer
38 views

E[X|Y]=E[X] When X and Y are Independent — Proof from Book Question

I'm trying to understand a proof in a book for the following Theorem: Let $Y \in L^1 (\Omega, A, P)$ and suppose $X$ and $Y$ are independent. Then $E[Y|X]=E[Y]$. Proof: Let $g$ be bounded Borel. ...
1
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1answer
22 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
1
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2answers
37 views

Problem with a probability question.

I'm having a problem with this probability question: 2 robbers disguises themselves as customers in a bank that has 38 customers (non-robbers). Police try to reveal the robbers by using a ...
3
votes
1answer
75 views

Probability of at least m in a row out of n? (generic formula)

In a previously asked question of mine, I was specific in asking for a 75% freethrow shooter, what is the probability he would make at least 5 freethrow shots in a row out of 10. That means he would ...
3
votes
1answer
45 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
1
vote
5answers
42 views

Flipping two coins, which is more probable?

When flipping two coins, is it more probable for the two coins to match (ex. Heads, Heads) or be different (ex. Heads, Tails).
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vote
1answer
42 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
0
votes
5answers
90 views

How to prove that the function $f(x)=0.1\,e^{-0.2|x|} $ is a probability density, and then use it?

So here's the integral, I'm having a hard time solving it. I even tried integration software, but it didn't help: $$ I=\int_{-\infty}^{+\infty}f(x)\,dx,\qquad f(x)=0.1\,e^{-0.2|x|} $$ The question ...
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0answers
23 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
2
votes
4answers
46 views

Very basic probability problem

Firstly, apologies if this question is too trivial or just plain inappropriate for this site... I want to catch a ball. I have exactly two chances to catch the ball. The ball is thrown in such a way ...
0
votes
1answer
20 views

Probability of picking from a sublattice

Short version The set of partitions of a four-element set forms a lattice. Suppose that I pick $n$ times from the set of tri- and bipartitions (i.e., the top element = quadripartition and the ...
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0answers
30 views

Coupon Collector's problem with possibility of failure

In the standard problem, there are n items and you need to collect all of them. I'm curious about what would be the expected number of times you need to draw if instead you also have a chance to draw ...
2
votes
3answers
34 views

How to find $E[\sqrt{X}]$ given only a distribution function?

I'm given a continuous distribution function $F(x)$ and I should assume that $X$ is some random variable. My goal is to find the expected value of $\sqrt{X}$. I'm very bad at probability but this is ...
0
votes
1answer
37 views

Do we need to use continuity correction if we use CLT to do normal approximation

In a hotel, large number of cups and saucers are washed everyday. The number of cups that are broken each day while washing averages $2.1$. The number of saucers broken each day averages $1.6$, ...
0
votes
1answer
19 views

Conditional probability new variable

Variables $X_1, ... , X_N, N$ are independent $X_i - \exp(1)$ and $N$ have geometric distribution with parameter $\frac{1}{2} $ . We have a new variable $Z= \min(X_1, X_2 , \ldots , X_N)$ I must ...