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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2
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0answers
21 views

Distribution of $f(x,|h|)$, being $|h|$ rayleigh distributed

INTRODUCTION Let's supose we receive the following signal: \begin{equation} y[n] = hx[n]+W[n] \end{equation} where: $x[n] = Ae^{j2 \pi f_c t}$ is the transmitted signal $f_c$ is the carrier ...
0
votes
1answer
31 views

confirming solution - elementary probability

Box 1 contains 1000 transistors, of which 100 are defective, and box 2 contains 2000 transistors, of which 100 are also defective. A box is taken at random and two transistors are drawn from it, at ...
2
votes
1answer
28 views

Max value of hypergeometric distribution?

I'm reading the book Probability Theory: The Logic of Science by Jaynes. While I'm reading chapter 3, on page 56, it says: Although the hypergeometric distribution $h(r)$ appears complicated, it ...
1
vote
1answer
23 views

construct confidence interval from proportions

Suppose you have a population of count data, i.e., $1,2,3, \dots, k$, you have a sample of the population of size $n$, and you have a confidence interval for the proportion of $1$'s , $2$'s,\dots$n$'s ...
-3
votes
0answers
38 views

How to compute P(Y>X)? [closed]

Compute $P(Y>X)$ when $F_{X,Y}(x,y)= \begin{cases} 3\over4& \text{if }(0< x < 2) \wedge (0 < y < 2x-x^2)\\ 0 &\text{elsewhere}\end{cases}$ $F_X(x)=\begin{cases}\frac32 ...
-2
votes
0answers
32 views

Show that $X_1,X_2, \ldots$ are identically distributed [closed]

Suppose $A_1,A_2,\ldots$ are i.i.d. with positive expected value and $$S_0=0, S_n=S_{n-1}+\min\{t>0: A_{S_{n-1}+1}+\cdots+A_{S_{n-1}+t}<0\}.$$ Let $\tau=\min\{n:S_n=\infty\}$. Let ...
1
vote
2answers
42 views

Confidence interval for Poisson distribution coefficient

This is an exam question, testing if water is bad - that is if a sample has more than 2000 E.coli in 100ml. We have taken $n$ samples denoted $X_i$, and model the samples as a Poisson distribution ...
0
votes
0answers
17 views

Median of n normal distributions

I have the following problem: I would like to know the distribution of $\mathrm{Med}[X1,\ldots,Xn]$, where $X1,\ldots,Xn$ are standard normal distributions $N(0,1)$. I know that one should consider ...
-1
votes
0answers
70 views

How to solve for X^2-2Yx+Y=0? [closed]

How can I solve for $x^2-2Yx+Y=0$? Note: Y is an exponentially distributed random variable with parameter lambda>0. The solution is the following: no real solution for $4Y^2-4Y<0$, so when ...
-1
votes
2answers
57 views

Number of tosses needed to determine if a coin is biased [closed]

I toss a coin 3 times to get heads every time. What is the minimum no. of tosses needed to determine with 99% confidence, that this coin is biased?
-1
votes
1answer
27 views

Calculate probability of meeting. [duplicate]

My friend and I decided to meet between 1 and 3 PM today. There is a condition that whoever arrives first will not wait for the other for more than 20 minutes. what is the probability that we'll meet ...
0
votes
1answer
33 views

Simulating Random Vectors Based on Conditioning

I'm working on a project where I need to simulate random vectors $(Y, X_1,\dots,X_n)$ in order to understand the joint distribution $f(y,x_1,\dots,x_n)$. I wish to simulate enough random vectors so ...
1
vote
1answer
31 views

Binomial Distribution word problem (basketball)

I made this problem up: Lebron James has a free throw percentage of 0.71 if he shoots 100 shots from the free throw line, what is the probability he will make 70 of those shots?. I used the binomial ...
0
votes
2answers
88 views

A mathematical brain teaser in probability field

That is a brain teaser but it is also a mathematical problem in probability field. One day, a man have been trapped into a building which has 18 floors. Now, he want to leave the building but there ...
0
votes
0answers
30 views

Elevator probability revisited

An elevator in a building starts with 16 passengers and stops at 4 floors. If each passenger is equally likely to get off on any floor and all the passengers leave independently of each other, what is ...
2
votes
4answers
84 views

What does this definition mean: $F_Y(y) =P(Y<y)$?

I am doing calculations on $F_Y(y) := P(Y<y)$, but I am clueless as to what $P(Y<y)$ means. For instance the following question: Given function: $f_X(x)= 2\lambda x e^{-\lambda x^2}$ when $x ...
3
votes
2answers
80 views

Probability with changing number of marbles

Given a bag containing 20 marbles of 5 different colors in this configuration: 8x Blue 6x Red 3x Green 2x White 1x Black How would you determine the probability of picking a marble of a specific ...
2
votes
3answers
77 views

What is meant by $P(X = x)$?

What is meant by the statement $P(X = x) = \theta$? As in, what is its English translation? I'm assuming that $X$ is a random variable and $x$ is a member of its sample space. Is it just "the ...
2
votes
1answer
35 views

How do you proof that F is a distribution function, when x > 0

I hope that someone could help me solve this question of my textbook: Let F (x) = e^(−1/x) for x > 0 and F (x) = 0 for x ≤ 0. Is F a distribution function? If so, find its density function. How do ...
1
vote
1answer
36 views

Convergence in Probability for a Sequence of Random Variables

I am trying to solve the following: Let $\{X_n, n ≥ 1\}$ be a sequence of i.i.d. random variables with density $f(x) = e^{−(x−a)}$, for $x ≥ a$ and $f(x) =0$, for $x < a$. Set $Y_n = \min(X_1, ...
-4
votes
1answer
30 views

Mean and Variance of a Function of an Exponential distribution [closed]

The question is: I know that the mean and variance of X are $1/4$ and $1/16$, but how would you find it for $Y$? I thought of using a moment generating function, but am confused as to how to do so. ...
2
votes
0answers
39 views

The polynomial is dense in $L^2$ with non-lebesgue measure

Assume the function $u\to \mathbb E[e^{iuX}]$ is analytic in a nbhd of $0$ where $X$: $\Omega\to \mathbb R$ is a random variable. Now I want to conclude that the space of polynomial, denoted by ...
5
votes
1answer
77 views

Solitaire probability

I would like to know the exact probability of the following game. I start counting from one to 13 and do this totally four times. On each turn I say a number and turn a card from a deck. Ace is ...
0
votes
0answers
20 views

ELO in non-50%-chance-to-win games.

So ELO system has been implemented in many games like League of Legends, DOTA, CS and others, in which 50% of the players win and 50% lose, so, if we're not taking personal skill into account, every ...
2
votes
2answers
55 views

Probability Modem is Defective

A store has 80 modems in its inventory, 30 coming from Source A and the remainder from Source B. Of the modems from Source A, 20% are defective. Of the modems from Source B, 8% are defective. ...
0
votes
2answers
55 views

Convergence in Probability Proof

I am trying to show the following: Let $X_1, X_2, . . .$ be $U(0, 1)$-distributed random variables. Show that $max_{1\leq k\leq n}X_{k} \to 1$ as $n \to \infty$ in probability. I am not sure where ...
0
votes
4answers
28 views

what does the union of those 3 events imply

Let there be 3 events: A=a dish got broken B= electric product stopped working C= the car got broken Write the following event, D= at least 2 problem occurred. $D=(A\cap ...
1
vote
1answer
19 views

What is the relationship between poisson, gamma, and exponential distribution?

I'm having a hard time understanding the intuitive relationship between these three distributions. I thought that poisson is what you get when you sum n number of exponentially distributed variables, ...
0
votes
4answers
46 views

Probability of a dice game?

The question is as follows, A game consists of throwing 3 dice simultaneously. If number 1 or number 6 appears, the player wins $\$1$ else player loses $\$3$. After 9 games, what is the amount ...
0
votes
1answer
35 views

Prove that the chance of ever getting more heads than tails with an unfair coin is 1/(p-1) where the chance of getting a head with each toss is 1/p

Imagine a game scenario in which you toss a coin indefinitely until the cumulative number of heads exceeds the cumulative number of tails, upon which you stop. Given the general case that there is a ...
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0answers
41 views

What is the product of two independent random variables (as mentioned below)?

Let $X$ and $Y$ be two random variables with: $\begin{equation} f_{X}(x) = \begin{cases} e^{-\lambda T} & \text{if } x = 0;\\ \lambda T e^{-\lambda T(1-x)} & \text{if } 0 < x \leq ...
-1
votes
0answers
25 views

absolute deviation for binomial distribution [duplicate]

Let $X_{1},X_{2},...,X_{n}$ be independent Bernoulli trials being a $1$ (success) with probability $\frac{1}{2}$ and $0$ otherwise. Let $$X=\sum_{i=1}^{n} X_{i}$$ be the binomial random variable with ...
0
votes
2answers
62 views

Probability of selecting same numbers by players (not a pigeon hole problem)

There are $n$ players who play a game of selecting numbers from a range $k$ over a period of time and the rules of the game are as follows : The game is played for 1 minute and the players are only ...
0
votes
1answer
14 views

Probability of return with 7% error

I have a problem understanding the answer of the following problem: A recent audit by the IRS of the returns she prepared indicated that an error was made on 7% of the returns she prepared last ...
0
votes
0answers
17 views

Weak convergence of a double sequence of random variables

Consider two sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Let's assume $X_n\xrightarrow{D} X$, $Y_n \xrightarrow{D} Y$, and $\{X_n\}$ and $\{Y_n\}$ are independent of each other. It is ...
1
vote
2answers
22 views

Why do We Refer to the Denominator of Bayes' Theorem the “Marginal Probability”?

Consider the following characterization of Bayes' Theorem: Bayes' Theorem: Given some observed data $x$, the posterior probability that the paramater $\Theta$ has the value $\theta$ is $p(\theta \mid ...
2
votes
2answers
75 views

Probablity of $k$ balls go into at most (including) $m$ boxes, with $n$ given boxes ($m\le n\le k$)

Suppose I have $n$ boxes, and an infinity of balls. All balls are regarded the same. I randomly throw balls into the boxes, each time the ball may be thrown into one of the box, with equal ...
1
vote
0answers
18 views

How to solve for a Phase Function Cumulative Distribution function (CDF) calculation give a pdf …

I am attempting to solve for the CDF (more specifically the inverse CDF, but that is easy once I have the CDF) - Cumulative Distribution function given a Probability Distribution Function (pdf) and g ...
2
votes
0answers
15 views

What does the s-Transform (exponential transform) mean conceptually? What does it show us?

I don't understand the conceptual idea. If I have PDF, and I calculate its s-transform for some s, what do I know that I did not know before?
0
votes
1answer
28 views

limit of a product of independent random variables

I'm stuck with the following problem: Let $X_1, X_2,...$ be independent random variables which uniformly distributed on intervals $[-a_1,a_1],[-a_2,a_2],...$ respectively. Define ...
1
vote
2answers
34 views

Uniform distribution on $\{1,\dots,7\}$ from rolling a die [duplicate]

This was a job interview question someone asked about on Reddit. (Link) How can you generate a uniform distribution on $\{1,\dots,7\}$ using a fair die? Presumably, you are to do this by combining ...
2
votes
0answers
11 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
0
votes
1answer
28 views

Which distributions cannot be captured using this representation of the joint probability?

Reading through Probabilistic Graphical Models, I came across this definition along with an accompanying statement. Let $P(x_i) = \theta_i$. Define: $$P(x_1, \ldots , x_n) = ...
1
vote
1answer
36 views

Statistics- Finding Probability

A local lawn service has determined the average time it takes to mow an average residential yard is thirty-five minutes. If mowing times are independent and constant, what is the probability it will ...
0
votes
0answers
9 views

Finding the conditional probability of the supremum of a positive continuous martingale [duplicate]

I'm trying to prove that if $(M_t, F_t)$, $t\geq0$ is a positive continuous martingale which converges to zero almost surely, then for all $\alpha>0$, $$ P(\sup_{t\geq0}M_t\geq ...
2
votes
2answers
99 views

A casino owner is concerned based on past experience that his dice show 6 too often

Currently I am trying to understand hypothesis testing, but I have a problem with the following question: A casino owner is concerned based on past experience that his dice show 6 too often. He makes ...
1
vote
1answer
27 views

Probability of Limsup of a bunch of events

In "Probability Theory" by Athreya, I encountered the following question: Let $\{A_n\}_{n=1}^\infty$ be a sequence of events on a common probability space. Suppose $$\sum_{n=1}^\infty P(A_n\setminus ...
2
votes
2answers
73 views

Why is it impossible to hold these probability beliefs?

Why is it impossible to hold these probability beliefs? \begin{align*} P(a) & = 0.3 & P(a \land b) & = 0 \\ P(b) & = 0.4 & P(a \vee b) & = 0.8 \end{align*} I know that you ...
0
votes
1answer
32 views

what does E[$c^x$] mean in probability

Hello I'm self studying probability this summer and I would like your help to clarify me on this question. Let x be such that $P(X=1)=p=1-P(X=-1)$. Find $c≠1$ such that $E[c^x]=1$ Can anyone tell ...
-1
votes
1answer
26 views

conditional probabilities with Not

Example: find P(A|B), when P(B|A) = 0.8, P(B|~A) = 0.3, P(A) = 0.2 What can i do with P(B|~A), and does it look any different in the Venn diagram than P(~A)? I'm super confused.