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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1answer
8 views

Probability card game - 100 games

If a card game involves drawing a card from the deck, replacing it, shuffling, and then drawing another card (so each event is picking 2 cards). If the two cards are the same suit, then you win 2 ...
0
votes
0answers
29 views

Compute Var(x=X1+X2+…+Xn)

Compute $Var(X_1+X_2+...+X_n)$ given $X_1,X_2...$ are iid.,$EX=\mu,Var(X)=\sigma ^2$,and $Var(N)=\sigma [n]^2$, N is a random variable of nonnegative integers independent with X, and my solution ...
2
votes
1answer
317 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
3
votes
2answers
134 views

Maximum-Value Secretary Problem

Background: The classic secretary problem has the simple solution of rejecting the first 1/e applicants and then selecting anyone who was better than the best in the rejected set. However, in the ...
2
votes
0answers
16 views

How to show that $p(t|x,\mathbf x,\mathbf t)= \int p(t|x,\mathbf w)p(\mathbf w|\mathbf x, \mathbf t)d\mathbf w $

The following paragraph is approximately cited from Bishop's book, Pattern Recognition and Machine Learning. In carve fitting problem, we have training data $\mathbf x$ and $\mathbf t$, along ...
-1
votes
1answer
23 views

Probability of asymmetric random walk returning to the origin

Consider the random walk $S_n$ given by $ S_{n+1} = \left\{ \begin{array}{lr} S_n+2 & with & probability & p\\ S_n - 1 & with & probability & 1-p \end{array} ...
0
votes
1answer
436 views

Correlation between complex random variables

I am struggling to find the correlation between two complex r.vs; X and 1/Y i.e. E{X*/Y}, where '*' denotes the conjugation operator. The complex r.s X and Y are correlated with each other with known ...
-2
votes
0answers
28 views

I need help to solve this complex question [duplicate]

Peter has 12 pairs of socks and 6 pairs of gloves in different colors. His socks are in green, yellow, black, and grey (3 pairs each). Peter's gloves are either blue, black, or red (2 pairs each). ...
0
votes
2answers
31 views

Tricky Cardinality Question/Riddle

Mike is a child and he is playing with lego box. He begins by counting the number of pieces in his lego box. He recognize that some pieces are green, some pieces are blue, and some are green and blue. ...
0
votes
0answers
16 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
-1
votes
2answers
34 views

Can anybody help me to solve this counting problem [on hold]

A color on screens is the result of a combination of three colors red (r), blue (b), and green (g). A color c can be represented by the formula $$c = p_rr + p_bb + p_gg$$ where $$0 ≤ p_r, p_b, p_g ...
4
votes
8answers
3k views

Expected value of sums

Suppose we draw cards out of a deck without replacement. How many cards do we expect to draw out before we get an ace?
0
votes
0answers
5 views

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
0
votes
1answer
52 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
2
votes
1answer
40 views

How many numbers must be selected from 100…999 so that three of them have the same sum of digits?

A box contains 900 cards enumerated from 100 to 999 (Each number appears once and just in one card). I took some random cards without looking at them and calculated the sum of the digits in each one. ...
0
votes
0answers
32 views

Why use the expected value of $y$, $E(y)$, in simple linear regression.

I am learning about linear regression and I have ran into a bit of confusion. I'm trying to relate what I've learned in my probability and mathematical statistics course (in particular, expected ...
0
votes
0answers
9 views

Using an “auxiliary random experiment” to achieve a desired significance level

My question is somewhat simple, but, nonetheless I am not entirely convinced I am solving it correctly. I need to use the use the Neyman Pearson Lemma to test for $H_o : \theta = .5$ vs. $H_1 : ...
7
votes
2answers
85 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
2
votes
1answer
15 views

Uniformly Distributed random varibles

Question:Suppose $X$ is a uniformly distributed random variable with possible values $1,2, \ldots, 10$. Compute the expected value and variance of $X$. I have started with making a column ($x$ on the ...
-1
votes
0answers
36 views

How calculate the probability that there is a row in which there is no silver coin?

There are $n ^ 2$ coins, and $n$ of them are made of silver. The coins are set at random in $n$ rows, with each row having $n$ coins. How do we calculate the probability that there is a row in which ...
-4
votes
1answer
17 views

Find the 95% confidence interval and interpret the results [on hold]

A random sample of 38 200-meter swims has a mean of 3.96 minutes and the population standard deviation is 0.06 minutes. Construct a 95% confidence interval for the population mean time. Interpret the ...
1
vote
1answer
15 views

Expected value for sum of iid normal variables squared

Let $X_i$ be iid from a $N(\alpha, \alpha)$ distribution. I am trying to find $E[\sum_1^n X_i ^2]$ and thought that I would be able to transform the statistic $\sum_1^n X_i ^2$ into a chi-squared ...
1
vote
1answer
45 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
1
vote
1answer
14 views

Covariance and Correlation in Multinormal random variable

Find the covariance and correlation of $N_i$ and $N_j$, where $N_1, N_2, \ldots,N_r$ are multinormal random variable. At the beginning, I think that I have: ...
1
vote
0answers
23 views

Joint Probability Question

I have a question regarding join distributions. For this question, I have to find the probability that P(X+Y=0). I've attempted multiple different ways to solve this problem, but I keep getting 0 as ...
1
vote
1answer
21 views

Random variable and distribution - number of tests a teacher has to make

$100$ students do a test. The probability of failing the test is $0.6$, those that failed, do a retest, the probability of failing the retest is $0.5$. Those that fail the retest do another retest. ...
0
votes
0answers
7 views

Lists of common sufficient statistics

Can someone suggest a source for common sufficient statistics for exponential families? For example, I'm looking for something more comprehensive than the Wikipedia page for sufficient statistics, ...
0
votes
0answers
10 views

Rationale behind formula relating death probability and observed mortality rate

With $M_{x,t}$ stands for the time-$t$ observed mortality rate between ages $x$ and $x+1$ (formulas given below) and $q_{x,t}$ the time-$t$ death probability between ages $x$ and $x+1$ (the ...
0
votes
0answers
19 views

Return probability in markov chain

Given the following markov chain : $p_{0,1}=1$ (if we are in state 0, we must go to state 1) $p_{i,i+1}=p_{i,i-1}=0.5$ There are infinite (countable) states. I defined $T=inf\{n>0 : X_n=0 | X_0 ...
0
votes
1answer
15 views

Reason behind convergence in probability definition

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$ $$\lim_{n\to\infty}\Pr\big(|X_n-X| > \epsilon\big) = 0$$ But ...
0
votes
0answers
13 views

pdf for the sum of squared iid normal random variables

I am trying to find the distribution/pdf for the sum of squared $X_i$ iid observations from the normal distribution $X_1 ,..., X_n$ ~ $N(\alpha , \alpha)$, i.e. $X_1 ^2 + X_2 ^2 +...+ X_n ^2$. I was ...
0
votes
0answers
14 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
2
votes
1answer
22 views

Combinatorics/Probability unordered lists

I don't really understand these unordered lists problems such as... Q: John goes to a store and buys 10 pieces of fruit from the selection of apples, bananas,peaches and pears at random. What is the ...
0
votes
1answer
28 views

Expectations of squared sum question

I can't seem to figure out why these expectations turn out the way they do, I am currently studying about the Fisher Information. If $X_1,X_2,...,X_n $ are all iid Poission($\lambda$) , then going ...
3
votes
0answers
16 views

On the probability of singular matrices containing whole numbers

Today in class - my teacher was teaching determinants . He gave us problems to solve of various kinds , including various row - column operations and determinants properties. But one thing that ...
0
votes
2answers
29 views

Probabilty exam question

I would like some help with what direction to take in this question.I find it difficult to decide what rule I need to use when I read a question. Cars pass at an average rate of 1 every 10 seconds. ...
0
votes
0answers
26 views

Drawing 6 balls of different colours

Hi I have an exam on Monday and am doing a few probability questions. I have checked the mark scheme for the answer to the following question however the method isn't stated. Could someone please ...
0
votes
1answer
78 views
+50

Result of a $2D$ random walk with position dependent probabilities of step

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk:- A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
5
votes
0answers
184 views
+100

A fun card game involving probability, getting all 13 ranks (any suit(s)) vs. 5 in a row of red or black.

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, shuffled well before each hand is drawn, and randomly draw cards from it one a time ...
1
vote
2answers
61 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
1
vote
1answer
40 views

Significance level for a hypothesis test for a linear regression

Consider linear regression model $Y_i=a+b\cdot x_i+\epsilon_i$, $i=1,2,3,4,5$, where $a,b\in\mathbb{R}$ are unknown and $x_1=x_2=1,x_3=3,x_4=x_5=5$, $\epsilon_i$ are iid, normally distributed with ...
2
votes
1answer
22 views

Conditional Probability of Sinking Ship Question

Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a ...
1
vote
2answers
19 views

Prove that markov chain is recurrent

I have the following markov chain : $S=\{0,1,2,3\}$ $p_{i,0} = q$ (if we are in one of the states $0,1,2,3$ we can return to $0$ with probability $q$) $p_{i,i+1} = 1-q , i\in\{0,1,2\}$ (if we are ...
1
vote
0answers
16 views

Compare 2014 to 1998, 2014 has a 90% chance of being warmer than 1998?

According to NASA, 2014 has a 38% chance of being the warmest year, 1998 has only a 4% chance of being the warmest year. 2014 or 1998 have a 42% chance of being the warmest year. Since I eliminated ...
2
votes
1answer
31 views

Average difference between two odd numbers of equal length

If I select two different odd numbers of binary length $l$, what is the formula that will tell me the average difference between those two numbers? Note that the high order digit must always be $1$, ...
0
votes
2answers
42 views

How to decide the randomness of a dataset by checking the prime numbers inside it?

So it is weekend! I am reading currently a book where I found this sentence: "71 percent of men said they had a 'good sense of direction'. Only 47 percent of women reported same thing.", and I thought ...
0
votes
0answers
10 views

Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
-1
votes
2answers
44 views

Conditional Probability Question. [on hold]

A letter is known to have come from either 'TATANAGAR' or 'CALCUTTA'. On the envelop just two letters 'TA' are visible. What is the probability that the letter has come from (i) TATANAGAR (ii) ...
0
votes
2answers
33 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
0
votes
1answer
22 views

Probability Distribution sampling problem

$\text{*The below problem was asked in geometric distribution section}$ In a population there are $50\%$ Male and $50\%$ Female What is the probability to find $2$ Females in a row out of $10$ ...