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
16 views

Destined pair 'guessing' game

n people participate in a game. Before the game the participants are put into random secret 'destined' pairs. Each round the participants pick1 their own pairs and ...
0
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0answers
15 views

probability,calculus

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
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votes
3answers
24 views

Is every bounded sequence of random variables in $L^1$ convergent? [on hold]

If $\{X_n\}_{n>0}$ is a bounded sequence of random variables is it true that $E(X_n)$ converges?
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votes
0answers
24 views

Should I use law of large numbers or Chebyshev inequality?

I think the answer is zero. Can anyone tell me whether I should use Weak Law of Large Numbers or Chebyshev inequality . I just need a hint how to proceed. Is my answer 0 correct? Thanks link to ...
0
votes
1answer
28 views

Need Help with continuous random variable probability problem [on hold]

Suppose that an electric device has a life length $X$ which is considered as random variable with pdf: $f(x)=e^x$, $x>0$. Suppose that the cost of manufacturing one such item is $2$. The ...
2
votes
0answers
39 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
0
votes
1answer
25 views

Continuous probability - calculate probability of r.v and distribution function

This is the question: $X$ is a continuous random variable whose probability density function is given by $$f(x)=\begin{cases} \frac{1}{9}x^2 & \text{if $0\leq x \leq 3$}.\\ 0 ...
1
vote
1answer
37 views

How to show a sequence of independent random variables do not almost surely converge by definition?

I have a sequence of independent random variables $X_1, X_2, \ldots$ where $$ X_n = \begin{cases} 1 & \quad \text{with probability} \ 1/n \\ 0 & \quad \text{with ...
1
vote
1answer
27 views

Door Prizes - Probability [on hold]

Joe goes to a party with three friends. There is a drawing for four door prizes. Each person has an equal chance of wining a prize. No one can win more than one prize. If there are totally thirty ...
0
votes
1answer
19 views

How to show convergence in probability by just using the definition?

I have a series of random variables $X_1, X_2, \ldots$ where $$ f(X_n) = \begin{cases} 1/n & \quad \text{if} \ X_n = 1 \\ 1-1/n & \quad \text{if} \ X_n = 0 \\ 0 & ...
2
votes
2answers
35 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
2
votes
1answer
34 views

What is the intuitive difference between almost sure convergence and convergence in probability?

It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
0
votes
1answer
19 views

How can I compute the mean of a sequence of random variables?

Suppose that I have a sequence of random variables where $X_1, X_2, \ldots$ where the pdf of $X_n$ is equal to: $$ f_n(x) = \begin{cases} (n-1)/2 & \quad \ -1/n < x < 1/n \\ ...
0
votes
1answer
27 views

Continuous Probability - Bus Arriving

I am trying to do the following question: Number 24 and number 42 buses arrive independently at the corner of Mayeld Road at a random rate of 3 and 4 per hour respectively. You arrive at the ...
3
votes
1answer
14 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
0
votes
0answers
19 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
0
votes
0answers
23 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
1
vote
0answers
36 views

Single, 6-sided die probability

I'm working on an assignment and I'm more or less new to stats. It might be the wording of the questions that's getting me as well. It deals with a regular 6-sided die. 1.a) What is the mean number ...
0
votes
0answers
20 views

poisson process(exercice)

Let $N_t$ a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1)Show $P(B_t \geq x \ and\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq u)du$ with ...
0
votes
1answer
13 views

Finding the MLE for an open interval.

So the problem says: Let $X = (X_{1},...,X_{n})$ be a random sample, where $X_{i} \sim Unif (0, \theta _{0})$, where $\theta _{0} \in (0,\infty)$ is unknown. Find the maximum likelihood estimator $T$ ...
0
votes
0answers
21 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
0
votes
0answers
11 views

Conditional probability,two conditions

A doctor operates on patient with a certain disease if he is 80% sure that he has it.We have a patient for whom the doctor is 60% sure that he has the disease,so he makes him do another test which is ...
0
votes
1answer
10 views

Test predictability with Bayes' Theorem

Say we have a disease and a test for it. P(A :=a person has the disease)= 0.01 ( example) P( B:=test is positive | A )=0.95 Is this enough information to calculate the probability that a person has ...
3
votes
1answer
44 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
0
votes
0answers
11 views

Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
0
votes
4answers
39 views

Probability of second card being an ace

I have this task about cards: Consider choosing a card from a well-shuffled standard deck of 52 playing cards. Suppose that, after the first extraction, the card is not reinserted in ...
1
vote
0answers
10 views

renewal process and Markov property

Let $A_t=t-S_{N_t -1}$ with $N_t$ a renewal process 1)Show $A_t$ checks the Markov property my proof: $S_{N_t}=X_1+\cdots+X_{N_t}$ and the increments are independents $$P(S_{N_t-1}=t-y\mid ...
1
vote
0answers
13 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
2
votes
1answer
12 views

Inverse of Gaussian CDF, Sum

Consider the following setting. Let $k = 1, \ldots, n$ and define $$y_k= \Phi^{-1}\left(\frac{k}{n+1}\right),$$ where $\Phi$ is the inverse of the CDF of a standard normal. I noticed numerically ...
1
vote
1answer
45 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
0
votes
1answer
48 views

The normal distribution - how to calculate the integral

Question: It was found that the mean length of $100$ parts produced by a lathe was $20.05$ mm with a standard deviation of $0.02$ mm. Find the probability that a part selected at random would ...
3
votes
1answer
19 views

Probability of working machine with $3$ components

I have this task to do: A machine is composed of $3$ components, which function independently of each other with probabilities $p_1$, $p_2$ and $p_3$, respectively. Calculate the probability ...
0
votes
0answers
41 views

Finding $P(A \cap D)$ when $P(A), P(B), P(C), P(D), P(A \cap B), P(A \cap D) + P(B \cap C)$ is known

If: $$P(A) = 0.26, P(B) = 0.39, P(C) = 0.15, P(D) = 0.20$$ $$P(A \cap B) = 0.21$$ $$P(A \cap D) + P(B \cap C) = 0.11$$ and $$P(A \cap C) = 0$$ $$P(B \cap D) = 0$$ Is it possible to find or make an ...
-1
votes
0answers
21 views

characterizing the PDF of the function of i.i.d. Gamma variables [on hold]

everyone, I am facing the following problem. I will be very appreciated if anyone can help me on this. $h_i$ and $g_i$ ($i \in \left\{ {0,1,2,...,N} \right\}$) are i.i.d. Gamma variables, the PDF of ...
0
votes
0answers
16 views

What is the probability of unions of intersections?

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
0
votes
0answers
23 views

Expected value and standard deviation for infinite sample with probability

Problem: A recruiting firm finds that $20$% of the applicants for a particular sales position are fluent in both English and Spanish. Applicants are selected at random from the pool and interviewed ...
4
votes
1answer
310 views

Is there a way to find expected value of equation?

If the random variable $X$ is binomially distributed with parameters $n=6$ and $p=0.3$, what is $$E(4+3X^2)$$ I know $E(X) = np = 1.8$. I solved this problem by finding $P(X)$ of all $X$ using ...
1
vote
1answer
32 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
0
votes
0answers
10 views

independence of chi square distributions

We already knew that if two independent chi-squared random variables, then their sum is also chi-squared with the degree of freedoms is the sum of theirs. How about the converse? If $X\sim\chi^2(n)$ ...
0
votes
1answer
43 views

What is the probability that the sum of two successive outcomes is 5? [on hold]

I roll a single die repeatedly. The game stops once two successive sum is 5 or 7. What is the probability that the game stops at the sum of 5?
0
votes
1answer
50 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
0
votes
1answer
26 views

If $Y = (\mathcal{N}(\mu_1,\sigma_1^2) + \mathcal{N}(\mu_2,\sigma_2^2))^2$, what is $\Pr(Y>\mathrm{E}[Y])$?

Given $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, with $X_1$ independent of $X_2$, as well as $Y = (X_1 + X_2)^2$, what is $\Pr(Y>\mathrm{E}[Y])$? ...
0
votes
1answer
19 views

Finding a Conditional Probability

This is the given info: $P(A)=2/5$ , $P(A \cup B)=3/5$ , $P(B|A)=1/4$ , $P(C|B)=1/3$ , $P(C|A \cap B)=1/2$ , Find $P(A|B \cap C)$ I know that $P(A|B \cap C)$ is equivalent to $P(A ...
0
votes
1answer
24 views

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ , what is the mean of $X$?

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$ and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is $\frac 12$ $\frac 1{\sqrt2}$ $\frac 13$ ...
1
vote
2answers
28 views

Obtaining probability density function $f_Y(y)$ when we know joint probability distribution $f(x,y) = 1/(x+1)$

Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by $$f_Y(y) = \int_{y-1}^1 ...
-3
votes
0answers
21 views

If $X$, $Y$ are random variables such that $E(X\mid Y=y)$ is constant for all $y$, then show that $E(XY)=E(X)E(Y)$ [i.e.,$\text{Cov}(X,Y)=0$] [on hold]

If $X, Y$ are random variables such that $E(X\mid Y=y)$ is constant for all $y,$ then show that $$E(XY)=E(X)E(Y)\qquad \text{[i.e. Cov}(X,Y)=0]$$
0
votes
2answers
44 views

Defining the states when we roll one single die repeatedly

We roll a single die and the game stops as soon as the sum of two successive rolls is either 5 or 7. We want to find the probability that the game stops at a sum of 5. It seems like Markov ...
-1
votes
1answer
41 views

Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution. Edit:- ...
1
vote
2answers
10 views

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer? $r_1 = r_2$. $r_1 = 10r_2$ ...
3
votes
1answer
15 views

Multi-stage Probability

I think the easiest way to explain what I'm having trouble with is to give an example question: A monkey is given 12 blocks: 3 Squares, 3 Rectangles, 3 Triangles, 3 Circles. Calculate the probability ...