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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0
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2answers
8 views

The skewness coefficient for given pdf?

$$f(x)= \begin{cases} f(x)=0.5-\frac{x}{8},& 0\le x \le 4 \\ 0, & otherwise \end{cases} $$ I have found $E(X) = \frac{4}{3}$ and $Var(X) = \frac{8}{9}$ Th problem says that the skewness ...
0
votes
1answer
10 views

Expectation of scaled sum of squares of iid random variables

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. What is ...
0
votes
2answers
37 views

Showing E$|X_n - X|\ge εP(|X_n-X| \ge ε)$, for $ε\ge0$.

If $X$ and $X_n$ are randoms variables, and $n\ge 1$, on a probability space, how do you show E$|X_n - X|\ge εP(|X_n-X| \ge ε)$, for $ε\ge0$?
-1
votes
1answer
29 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
2answers
53 views

Show that a random variable $T_x$ is uniformly distributed given that $T$ is uniformly distributed?

We have a lifetime $T$, which is uniformly distributed over $(0,b)$. We then introduce a new r.v., $T_x=T-x$, which is defined on $0<x<b$. We want to show that given $T>x$, the variable ...
-1
votes
1answer
87 views

Covariance between $1$ continuous and $1$ dummy variable

Suppose $Y$ is a continuous variable and $Z$ is a dummy variable that equals $1$ with probability $p$. $$\operatorname{Cov} (Z,Y) = p(1-p) {\left(E[Y\mid Z=1] -E[Y\mid Z=0]\right)}$$ How is this ...
1
vote
2answers
120 views

How to I find the distribution of $\log p(X)$ when $X$ is distributed under $p$?

I have a feeling there's no general solution to this problem, but I'll ask anyway. I have a multivariate PDF $p$ and, given a random vector $X\sim p$, I'd like to find the the PDF of $\log p(X)$. ...
-1
votes
1answer
38 views

Independence of intersection of events [on hold]

I have the events $A,B,C,D$ are mutually independent such that $P (A \cap B \cap C \cap D) = P(A)P(B)P(C)P(D)$ and I want to prove events $(A \cap B)$ and $(C \cap D)$ are also mutually independent ...
2
votes
1answer
44 views

Find a continuous PDF on $[0,6]$ for given probabilities

Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$. After some calculations I came up with ...
-4
votes
0answers
24 views

statistical probablity [on hold]

A man tosses two fair dice. What is the conditional probability that the sum of the two dice will be 7, give that (i) The sum is odd, (ii) the sum is greater than 6, (iii) the two dice had the same ...
2
votes
2answers
23 views

Pairwise independence vs independence

Two fair dice are thrown. We have three events: A: The first die shows an odd number B: The second die shows an even number C: Both are odd or both are ven Show that $A,B,C$ are ...
1
vote
0answers
16 views

Finding cdf, percentile, variance, and standard deviation from pdf.

$$f(x) = \begin{cases} 2(1-\frac{1}{x^2}) & \text{if }1\le x\le2 \\ 0 & \text{otherwise} \end{cases} $$ Compute the CDF of X: $$ \int^X_12(1-\frac{1}{y^2})dy = 2x+\frac{2}{X}-4 $$ So I ...
1
vote
0answers
14 views

Borel isomorphism between polish spaces

In my lecture on stochastics the following result has been used: For any uncountable Polish space $X$ there is a Borel isomorphism between this space and the real line. I was not able to find a ...
0
votes
1answer
25 views

Conditional entropy and independent conditioning variables

Let $X,Y,Z,Y',Z'$ be random variables where $Y\sim Y', Z\sim Z'$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are, in the sense that we have $p(X,Y,Z)=p(X|Y,Z)p(Y)p(Z)$ ...
0
votes
0answers
11 views

Ehrenfest chain

In the Ehrenfest model,Let Xn denotes the number of balls in the left urn. And there are N balls total. When we calculate P(Xn+1=i+1|Xn=i, Xn-1=in-1,...,X0=i0), why don't we taking account of the ...
0
votes
0answers
9 views

Finding recurrent states given a Markov chain

I have trouble in approaching the problem where: Consider a Markov chain $X_n$ , $n ≥ 0$, with state space $S = N = {0, 1, ...}$ and transition function $$ p(x,y) = 1/7, y=0 $$ $$ p(x,y) = 2/7, y∈ ...
29
votes
13answers
8k views

% of % - Please Help Me Prove My Friend Wrong

Here is the situation: My friend and I are at an impasse. I believe I'm correct, but he's so damn stubborn he won't believe me. Also, I'm not the most articulate at explaining things. Hopefully ...
1
vote
1answer
26 views

I roll a fair die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears.

I roll fair a die repeatedly until I get $6$, what is the probability that neither $1$ nor $2$ occurs before $6$ appears. Not sure how to go about this.
3
votes
3answers
79 views

Conceptual Statistics. Define for this problem, population, Samples and Estimators, and when is Normal Dist?

Students in Stanford are supposed to spend on average 3 hours of time per week for every credit hour they take. Last year, 263 randomly selected seniors were contacted and asked how much total time ...
1
vote
3answers
23 views

Probability of an even number of sixes

We throw a fair die $n$ times, show that the probability that there are an even number of sixes is $\frac{1}{2}[1+(\frac{2}{3})^n]$. For the purpose of this question, 0 is even. I tried doing ...
-2
votes
0answers
14 views

Poisson Counting Insurancee example [on hold]

An insurance company finds that for a certain group of insured driver , the number of accidents over each 24 hours period rises from midnight to noon and then declines until the following ...
0
votes
1answer
30 views

How do you picture: $\Pr(B|A)$ shrunk down by $\Pr(A)$?

Though understanding these diagrams, I do not understand how to visualise the following explanation: $\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it ...
2
votes
1answer
18 views

Probability: Finding the Number of Apples Given Two Scenarios

You have a bag containing 20 apples, 10 oranges, and an unknown number of pears. If the probability that you select 2 apples and 2 oranges is equal to the probability that you select 1 apple, 1 ...
-1
votes
2answers
28 views
5
votes
1answer
49 views

How to calculate the shortest interval, for $P ( X ≤ 1 . 645) = 0 . 95$?

The problem statement said: Based on the fact that $\Phi(1 . 645) = 0 . 95$ find an interval in which $X$ will fall with $95\%$ probability. Therefore: Since $P ( X ≤ 1 . 645) = 0 . 95, ( -∞ , ...
1
vote
1answer
12 views

Probability density function in Rayleigh distribution

It says that $$ f(x;\theta) = (x/\theta)e^{-x^2/(2\theta^2)}, x>0 $$ is the Rayleigh distribution. And asks to verify that $f(x;\theta)$ is a legitimate pdf. Can you explain how to verify ...
4
votes
1answer
32 views

How many ways can we deal a 13-card hand with at least one suit that does not appear?

In dealing a $13$-hand card that with at least $1$ suit that does not appear, I came up with this: We can choose $3$ of the $4$ suits, as in $3 \choose 4$, and then $13$ cards out of the $39$ cards ...
0
votes
1answer
37 views

Conditional Probability Problem: Two Radios from Two Factories

Q: There are two local factories that produce radios. Each radio produced at factory A is defective with probability .05, whereas each one produced at factory B is defective with probability .01. ...
0
votes
1answer
28 views

Find $c=c(n)$ so $T = c \sum_{i=1}^{n} |X_{i}|$ is an unbiased estimator.

I'm having some trouble trying to solve the following problem: Assuming that $X =(X_{1},\ldots,X_{n})$ is a random sample from the normal distribution with mean $0$ and unknown standard deviation ...
0
votes
1answer
20 views

Moments of a random sum with bounds Poisson distributed?

We have that $N$ and ${X_1,X_2,\dots}$ are all independent and that $f(x)=Cx^2(1-x)^2$. Then, we have: $$Z=\sum_{j=1}^{N+1}X_j$$ $N$~Poisson$\lambda$. Find the expectation and the variance of $Z$. ...
1
vote
2answers
32 views

Does the sum of Poisson random variables have a Poisson distribution?

So I have been taught that the sum of Poisson random variables have a passion distribution. However, I have a problem with this. Suppose you have a Poisson random variable $X$ with $E(X) = a$. Then ...
0
votes
1answer
21 views

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers. [on hold]

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers $Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
0
votes
2answers
30 views

How does one find the mode of a distribution without counting manually?

I know if I have a set of elements $\lbrace 1,2,3,4,4,4,5,8,9\rbrace$ Then the mode is $4$ in this case. How do I find the mode for more complex distributions? I have formulas that give me ...
0
votes
1answer
9 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
0
votes
1answer
403 views

Simple Probability Matrix

Question: Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will ...
1
vote
4answers
62 views

$2$ players take turns and draw from a box containing $1000$ balls, $3$ of them are black.

I'm not sure how to tackle this question. Assume a box containing $1000$ balls, $3$ of them are black and the rest are white. $2$ players $A_1$ & $A_2$ take turns and draw from the box without ...
2
votes
0answers
28 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
0
votes
1answer
26 views

Use Maximum Likelihood Estimation to guess which dice got selected

We have two six-sided dice (faces numbered 1 through 6) and two tetrahedral dice (faces numbered 1 through 4). Someone selects two of them and throws each once. Then they tell us the sum of the ...
2
votes
1answer
103 views
+50

Exploding dice in a dice pool

Say we role $n$ identical, fair dice with $d$ sides (every side comes up with the same probability $\frac{1}{d}$). On each die, the sides numbered from $1$ to $d$ with no repeating numbers on any one ...
1
vote
1answer
768 views

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms…

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a) What is the ...
-1
votes
0answers
32 views

show that $Y_1$ is unbiased for $\theta$ and find its variance [on hold]

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{$P_0$}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. Show ...
0
votes
0answers
24 views

Probability of at most $K$ consecutive zeroes in a sequence of 0s and 1s.

I am studying for a probability in computer science course and came upon this exercise problem that I have trouble solving. I need this to prove that in a sequence W of length n, consisting of 1s and ...
0
votes
1answer
385 views

partial differentiation of function of expectation of random variable

We have $E(U)=\int_0^B V f(V) dV + B \int_B^\infty f(V)dV$; Here $V$ is random variable. $E(U)$ stands for expectation of $U$. We have $Z=f(E(U))$ i.e. $Z$ is function of $E(U)$. Can we write ...
0
votes
1answer
19 views

Urn with white and black balls, random variable, conditional probability

An urn contains white and black balls with $p_w=p$ and $p_b=1−p$. If I made some extractions with replacement, what are the support and the probability function of $X_a$, where $X_a$ is the random ...
0
votes
1answer
22 views

Calculate characteristic function

$p(n)=(1-r)^2nr^{n-1},n=1,2,...$ $f(z)=1/(1-z)$ has derivative $f'(z)$ with convergent power series $f'(z)=1/(1-z)^2=1+2z+3z^2+...$ the answer I have got is $(1-r)^2e^{it}(1-re^{it})^{-2}$ , I am not ...
0
votes
0answers
29 views

Probability Mass Function of a Sentence

We have a sentence: Some dogs are brown. We choose one letter (out of the 16) at random. Let Y be the length of the whole word containing the letter. How can I find the probability mass function of Y? ...
0
votes
1answer
55 views

The probability of the sum of $10$ dice rolls adding up to $57$

So the question is: given that you roll $10$ dice, what is the probability of the sum of the total dice rolls adding up to $57$? I know that there are three ways to do this: Seven die rolls must ...
1
vote
0answers
27 views

Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts $(2,3,5)$. So if you buy a ticket you can win a pot which will payout your ticket price multiplied by one of those numbers. The organizer expects a margin profit ...
2
votes
2answers
34 views

Average number of events happening if each happens with $p=\frac{1}{n}$ and we run it $10000 n$ times.

Let an event $e$ have probability of happening $\frac{1}{n}$. Let us assume we have $m$ independent possibilities for similar events to happen. With $m>>n$. What is the average number of times ...
-1
votes
1answer
25 views

Given probability distribution $f(x)=2-bx$ find $b$ and range for $x$

Suppose that the distances between houses and the center of a city are distributed with the density function: $f(x)=2-bx$, where $x$ denotes distance. If this is a proper density function, what can we ...