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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0answers
22 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
-2
votes
1answer
23 views

Find probability mass function from text [on hold]

$5$ persons (each independent of the other) when in a good mood it opens the tap with probability $\frac{1}{2}$ or in a bad mood with probability $\frac{1}{2}$. When that person is in a good mood it ...
1
vote
2answers
38 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
2
votes
0answers
64 views

Why is this the solution?

I have this exercise with the solution. Let $X\sim N(0,1)$. Show that $P(2X = 3Y + 1) = 0$ if $Y\sim \text{Poisson}(\lambda)$. I have this solution $P(2X=3Y+1)= P(\bigcup_{k=0}^{\infty}(2X = 3Y+1, ...
3
votes
1answer
57 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
0
votes
1answer
35 views

Joint probability density for independent variables

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = e^{-x_i}$, for $x_i > 0$ for $i = 1,2$ (a) Find the joint probability density function ...
-1
votes
0answers
33 views

Not understanding the results of standard deviation [on hold]

There are two scores from one sample from two measurements: $A_1=15.4$ and $A_2=16.6$ the standard deviation (in the range +/-1 $\sigma$) for the first ($A_1$) is 1.91, and for the second ($A_2$) is ...
0
votes
1answer
46 views

A prob =a random variable

When I read some proofs, some authors conclude that $P(A)=I_{A}$, where $A$ is an event and $I$ is the indicator function. They mean that $P(A)$ can take either $0$ or $1$. However, I do not ...
0
votes
2answers
37 views

How to find E(Y) given that the random variable X is exponentially distributed with lambda equal to 0.5?

Random variable $X$ is exponentially distributed with the parameter $\lambda$ equal to $0.5$. Define also $Y = 1 - 2X$ Find $E(Y)$ , Var(Y) and the moment generating function of Y. I have $f_x(X)= ...
-2
votes
0answers
37 views

Math test: Probability [closed]

I have participated into a university course: Basics of statistics and probability. Online test with one specific question is giving headache: "4 out of 8 servers are required to provide cloud ...
2
votes
1answer
48 views

Convergence of a sequence

Let $X$ be a random variable with a distribution function such that $n^t P(|X|>n) \to 0$ as $n \to \infty$, for some $t>0$. Then, I know that for any $\epsilon>0$, there exists $n_0\in ...
0
votes
1answer
41 views

Showing Convergence in Distribution for Conditional Random Variable

I am trying to prove the following: Let $X$ and $Y$ be random variables such that $Y | X = x$ ~ $N(0, x)$ with $X$ ~ $Po(\lambda$). Show that $\frac{Y}{\sqrt{\lambda}} \to N(0,1)$ in distribution as ...
-3
votes
0answers
34 views

Can someone show me how to answer this question? please [closed]

Your socks drawer contains 10 pairs of white socks and 10 pairs of black socks. Suppose you are in the dark and need to get one sock at a time, without knowing which one. How many socks do you need to ...
0
votes
1answer
59 views

How many tickets sold so that $2$ people won the first prize in a lottery?

In some lottery one can buy a ticket by choosing seven distinct numbers each of them from numbers ${\{1, 2, \dots, 45}\}$ (so $1/(45379620)$ is the probability to win the first prize). Every week the ...
3
votes
1answer
40 views

Computing Conditional Characteristic Function

I am trying to compute the characteristic function of the following: Let $X$ and $Y$ be random variables such that $Y\mid X = x\sim N(0, x)$ with $X\sim\mathrm{Po}(\lambda)$. Find the characteristic ...
3
votes
1answer
42 views

Probability of True Positive of a random variable defined by an integral expression

$\newcommand{\Prob}{\operatorname{Prob}}$Let's assume that we have a random variable with the following pdf: \begin{equation} f_T(x) = \int_0^\infty f_T(x,g) \cdot f_{g}(g) \, dg = \int_0^\infty ...
4
votes
1answer
56 views

Probability of consecutive floors on an elevator with more people

Another user posted this question about elevator occupants, which made me curious about a harder question. In a $t$-story building (with no basement), $n$ people get on an elevator on the first ...
0
votes
1answer
25 views

Elementary probability question involving a 4-sided dice rolled twice

I'm beginning some probability courses so please explain your reasoning as if I were stupid. We have a 4-sided dice. Our experiment consists of rolling the dice twice: Let event $A = \{$maximum of ...
-1
votes
0answers
109 views

Find measure such that…

I've a very concrete problem I can't solve. Consider the following function $k: [0,1]^2 \to \mathbb{R}:$ $$ k(x,y)=\begin{cases} 1 &\text{if } y > x \\ -1 &\text{if } x- \frac{1}{2} < ...
2
votes
3answers
251 views

Probability that after 10,000 steps (+-1) you'll end up at the origin. How to use Central Limit Theorem?

Starting at the origin and taking one step left or right with equal probability, what is the probability that you'll end up at 0 after 10,000 steps? I figured it'd be ...
3
votes
2answers
60 views

Probability that all colors are chosen

A box contains $5$ white, $4$ red, and $8$ blue balls. You randomly select $6$ balls, without replacement, what is the probability that all three colours are present. Most similar problems ask for ...
-2
votes
1answer
50 views

Let X and Y be two random variables with joint probability density function [closed]

$f(x,y) = k(1+xy)$, $0<x<1$ and $0<y<1 $ (a) Find the value of $k$ such that $f(x,y)$ is a valid joint probability mass function. (b) Are $X$ and $Y$ independent? Justify your answer. ...
1
vote
1answer
128 views

$X_1$, $X_2$ i.i.d., prove that $E(X_1\mid X_1+X_2) = E(X_2\mid X_1+X_2)$

I got to the point where I only need to prove that for every $h$ Borel, $$E( h(X_1+X_2) (X_1-X_2) ) = 0$$ This is obvious when $h$ is the identity function, but I don't know what to do. Thanks!
-3
votes
1answer
38 views

Find the expected value of a game [closed]

It costs $\$10$ to play a game. You have a $15\%$ chance of winning. You collect $\$30$ if you win. Otherwise you lose your $\$10$. Find the expected value.
2
votes
1answer
40 views

How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian

I was tying to prove the following: Given $(X,Y)$ a centered gaussian vector in $\mathbb{R}^2$ with the following covariance matrix $$ \Sigma = \begin{bmatrix} \sigma^2_x & \sigma_{x,y} \\ ...
-1
votes
1answer
24 views

Probability of hitting numbers 1 - 12 on a single zero roulette wheel [closed]

there is a 1/3 chance of hitting nos. 1 - 12 on any given spin -- but what are the odds of hitting numbers 1 - 12 twice, three times, four times and five times in a row?
0
votes
0answers
16 views

Markov chain state reached earlier than other state

Consider a Markov chain with $S={1, 2, 3, 4}$ and transition Matrix: $P=\begin{bmatrix} 0 & 1/2 & 1/2 & 0 \\ 0 & 0 & 1/2 & 1/2 \\ 1/2 & 0 & 0 & 1/2 \\ 1/2 & ...
1
vote
0answers
22 views

Median of the mean/max of an iid sample of exponential variables

I would like to know if one can obtain a simple analytic expression of the median of $T_n$ defined by $$ T_n = \frac{\overline X_n}{X_{(n)}}, $$ where $\overline X_n$ is the empirical mean and ...
0
votes
1answer
42 views

How to find the expected cost of an exponential probability?

The length $X$ of of a call follows the exponential distribution with mean $2$ minutes. In dollars, the cost of of a call of $x$ minutes is $3x^2-6x+2$. Find the expected cost of a call? The addition ...
0
votes
0answers
25 views

How to compute $\mathbb{E}(X \cdot Y \ \textbf{1}_{X \cdot Y \leq n})$?

Let, $X$ and $Y$ be two independent random variables. Then, $\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$. Now let $\textbf{1}$ bew the indicator function. Is it true the next statement true? ...
5
votes
1answer
64 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where ...
1
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0answers
19 views

search on a split data structure

I have the following problem: Part 1: Lets say I have n items in a data structure and I want to search for them. I know that a subset of my data $r \cdot ...
0
votes
1answer
42 views

Given a CDF find the PDF

Let $$F(x) = 1 − \Bbb e ^{-x^3}; x > 0$$ be the cumulative distribution function of a continuous random variable $X$. (a) Find the probability density function of $X$. (b) Find the value of $c$ ...
0
votes
0answers
45 views

How to calculate $P(A_1)$?

Let the outcome of a random experiment is one of $A$, $B$ or $C$ events. Also Let $A$ (or $B$ or $C$) comprised of two events $A_1$ and $A_2$ (or $B_1$ and $B_2$ or $C_1$ and $C_2$, respectively.) ...
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0answers
34 views

$\sup B_t$ has the same distribution as $\sup C_t$ for two brownian motions $B_t, C_t$

Let $(B_t)_{t \ge 0}$ and $(C_t)_{t \ge 0}$ be two standardized brownian motions. Now why is $\sup_{t \ge 0} B_t$ distributed same as $\sup_{t \ge 0} C_t$? This is a result we assumed as trivial ...
0
votes
2answers
35 views

Let $X$ be a continuous variable with probability density function $kx(1-x)^2$ for $ 0<x<1$

Let $X$ be a continuous variable with probability density function $f(x)=kx(1−x)^2$ over $0< x <1$, zero otherwise. $(a)$ Find a value of $k$ so that $f(x)$ is a proper density. $(b)$ Find ...
5
votes
5answers
136 views

What is the probability of winning a best of $1, 3, 5, \cdots$ to infinity?

A shady casino organizes a simple game with rules that follow: 1 die is rolled. If it lands on an even, the house wins. If it lands on an odd, the player wins. However, if the player loses he may ...
-1
votes
3answers
56 views

What can we conclude if $A$ and $B$ are events such that $p(A) + p(B) - p(A \cap B) = p(A)$? [closed]

If $P(A)$ and $P(B)$ are two events such that $p(A)+p(B)- p(A \cap B) = p(A)$, then: $p(A / B)=1$ $p(B/A)=1$ $p(B/A)=0$ $p(A/B)=0$ Please help and thanks in advance.
1
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0answers
31 views

Why is this definition of the mesaure $\mu$ not properly?

Consider the measurable space $$ (\left\{a,b,c\right\}^{\mathbb{Z}},\mathfrak{A}), $$ where $\mathfrak{A}$ is the product topology on $\left\{a,b,c\right\}^{\mathbb{Z}}$ which is generated by the ...
1
vote
1answer
38 views

What is Skorohod's Represntation Theorem Saying?

From Wikipedia: Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. ...
5
votes
1answer
33 views

$W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$?

This is a sample question for the actuarial exam MFE. Let $Z(t)$ be a standard Brownian motion. Let $W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$? The only thing I know is Ito's Lemma. So I ...
0
votes
1answer
31 views

Simple Expected Value Of Continuous Variable Question [duplicate]

The normal expected value that I am used to is the following with $f(x)$ as the probability density function: $E[X] = \int_{-\infty}^\infty{xf(x) \, dx}$ My basic probability textbook is doing a ...
2
votes
1answer
69 views

Showing Convergence in Distribution of Continuous Function of Sums of R.V.s

I am trying to solve the following: Let $X_1, X_2, . . .$ be i.i.d. r.v.s with mean $\mu$ and positive, finite variance $\sigma^2$, and set $Sn = \sum_{k=1}^{n} X_k, n ≥ 1$. Suppose that $g$ is twice ...
3
votes
1answer
46 views

Proving a Variation of the the Central Limit Theorem

I am trying to prove the following: Let $X1, X2, . . .$ be positive, i.i.d. r.v.s with mean $\mu$ and finite variance $\sigma^2$, and let $S_n = \sum_{k=1}^{n} X_k$ , $n \ge 1$. Show that $\frac{S_n ...
0
votes
1answer
26 views

What is the probability of an item in a list being chosen if both the list and item have different probabilities?

This is kind of a programming question, but more so a mathematical question. I have two lists that contain various items inside of the lists. Each List has a rate of being chosen, then each item in ...
1
vote
1answer
33 views

Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
-1
votes
0answers
38 views

Impact of perturbation on the eigen-values of 3 diagonal matrix [closed]

Lets consider a 3-diagonal matrix as following: $$ A(i,i) = 2 $$ $$ A(i,i+1) = -1 $$ $$ A(i,i-1) = -1 $$ The eigen-values of this system is known easily. How eigen-values would change if we add ...
-2
votes
1answer
37 views

How to Calculate Probability in class math? [closed]

Suppose that $22\%$ of MATH students get an A grade, and $67\%$ of MATH students are from Berlin. The probability of being from Berlin or getting grade A is $75\%$. $S$ is from Berlin. Find ...
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votes
1answer
27 views
1
vote
1answer
38 views

Definition of a discrete random variable

Here is the defintion of discrete random variable from "An introduction to probability and statistics" by Rohatgi. Let $(\Omega,S,P)$ be a probability space. An random variable $X$ defined on this ...