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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3 views

PDF of Sum of Two Random Variables

$X$ and $Y$ are uniformly distributed on the unit disk. Thus, $f_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi}, & \text{if} ~ x^2+y^2 \leq 1,\\ 0, &\text{otherwise.}\end{cases}$ If $Z=X+Y$, find ...
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0answers
5 views

Acceptance Probability

10 Students are applying for postdoctoral job. The employer will only choose one candidate. Student Data 1 (PhD) 2 (MSc) 3 (PhD) 4 (BSc) 5 (BSc) 6 (MSc) 7 (MSc) 8 (BSc) 9 ...
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0answers
7 views

probability and applied statics 4

In cairo 30% of residents listen to the local fm radio . ten residents are chosen at random? a) state the distribution of the random variable b) find the smallest value of s so that P (x >or equal ...
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0answers
8 views

Find critical Value

A Basketball scout randomly selected 144 players and timed how long each player took to perform a certain drill.The times in this sample were distributed with a mean of 8 minutes. The population ...
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2answers
57 views

A problem of probability of ball drawing

Each of the five bags contains $a$ white balls and $b$ black balls.One ball is drawn from bag I and transferred to the second bag without noting the color of the ball,then a ball from second and ...
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1answer
50 views

Probability of voting

One hundred students are divided into two equal groups. Both groups vote (yes or no). Find the probability that the groups 1 and 2 both reach a majority. Students vote independently and the ...
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1answer
24 views

Poisson Process find P(N(u)=i | N(t)=n)

given $$u<t, i \leq n, $$ find $$P(N(u)=i | N(t)=n)$$ in terms of $N(t)$ I'm not really sure if it's applicable, but I found a theorem that pretty much says that the number of events occurring ...
2
votes
2answers
45 views

For $x \sim $ exp

$$E[X^2\mid X>1] = E[(X +1)^2]$$ I understand that $P(X>s+t \mid X>s)=P (X>t)$, however, I don't know how the statement above should be immediately obvious?
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0answers
14 views

Probability of events [on hold]

1) 5 cards are selected from a 52-card deck for a poker hand. a) How many simple events are in the sample space? b) A royal flush is a hand that contains the A,K,Q,J, and 10, all in the same suit. ...
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votes
1answer
10 views

Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
2
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3answers
43 views

What does “twice as likely” mean?

Once in a while I hear people say something like X is twice as likely as Y. What they usually mean is: $$p(X) = 2 \cdot p(Y)$$ and - in the context they refer to - they usually have $p(Y) < ...
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2answers
27 views

Number of necessary stickers to complete a sticker album

I have the following problem, and I was hoping you guys could help me solve it: Consider a set of $t$ unique, collectable stickers (that accounts for the universe of collectable stickers, i.e., ...
1
vote
1answer
12 views

application of the sampling distribution of x

the GBAs of all students enrolled at a large university have an approximately normal ditribution with a mean of 3.02 and a standard deviation of 0.29 ..find the probability that the mean GBA of a ...
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0answers
940 views

Distribution of the sum of the $q$th largest observations to the sum of total for a power-law.

Where $X_1, X_2, \ldots,X_n$ are sorted independents r.v.s, where we index and order in such a way that $X_i >X_{i-1}$, $i>1$ where all realizations follow the same Standard Pareto distribution ...
3
votes
3answers
82 views

the probability that a patient recovers from a rare blood disease

The probability that a patient recovers from a rare blood disease is 0.4.. if 15 people are known to have contracted this disease what is the probability that (a) at least 10 survive (b) from 3 to ...
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6answers
702 views

What is the probability that a leap year selected at random will contain 53 tuesdays?

A leap year consists of 366 days. A leap year consists of 52 weeks(i.e. 364 days) and 2 days.
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1answer
45 views

What is the answer [on hold]

Joe is 80 percent sure that his missing key is in one of the two pockets of his hanging jacket, being 40 percent certain it is in the left-hand pocket and 40 percent certain it is in the right hand ...
0
votes
1answer
20 views

Expectation of conditional event for throwing a fair dice

A fair die (with face numbered $1,\ \ldots\ ,6)$ is independently rolled repeatedly. Let $X$ denote the number of rolls till an even number is seen and let $Y$ denote the number of rolls till $3$ is ...
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1answer
41 views

Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
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1answer
33 views

Bounding second moment of entropy

Entropy is defined as $E(-\log(P(x))$. We know it is bounded by $\log(r)$ when $r$ is the size of alphabet. Defining the second moment as $E(\log^2(P(x))$, how to show it is bounded?
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0answers
32 views

Probability Theory (Hypergeometric) [on hold]

Suppose that a sample of size $n$ is to be chosen randomly without replacement from a basket containing $N$ balls, of which $m$ are red. Let $X$ be the number of red balls selected. Derive the ...
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1answer
25 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
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2answers
25 views

Pobability four letter problem [on hold]

Four different letters $a,b,c,d$ are written and put in the four different envelopes $A,B,C,D$. Find the probability distribution for number of letters put in correct envelopes.
6
votes
1answer
72 views

I pull $17$ balls out of a bag, and there are $13$ distinct colors in the sample. About how many colors are in the bag?

I have a bag filled with different colors of balls. My goal is to determine the number of distinct colors that in the bag, but I am limited to taking a small sample. From a sample of $N$ balls, I ...
0
votes
1answer
13 views

permutations and combination

How many different strings of lights can be created by placing 40 coloured lights on a horizontal string if 12 of them are red, 6 are blue, 14 are green and 8 are yellow? Assume that lights of the same ...
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1answer
9 views

Bridge hand Combination/Permutation

A Bridge hand consists of 13 cards from a deck of 52 cards. In how many ways can a (bridge) hand consisting of 5 spades(♠), 4 hearts(♥), 4 diamonds(♦) and 0 clubs(♣) be selected?
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1answer
27 views

Urn with $n$ balls, one special

An urn contains n balls, one of which is special. If k of these balls are withdrawn one at a time, with each selection being equally likely to be any of the balls that remain at the time, what is the ...
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0answers
5 views

deterministic limit of gaussian distribution

Let $a$ be a random variable over some set $A$, and let $\mathcal A \subseteq A$ be an event. Let $\mathcal E \subset \mathbb R^n$ be another event, and let $x_1, \dots, x_n$ be several Gaussian ...
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0answers
9 views

probability problem about Use the abstract properties of probability measures

Let $X$ be an arbitrary random variable on a countable probability space with probability measure $P$: Use the abstract properties of probability measures and expectations to prove the following: (a) ...
0
votes
1answer
19 views

Probability Discrete Math

{1,2,3,4,5,6,7,8,9} What is the probability that the sum of any of these three numbers is odd? I know that I should use $ n \choose k $ somehow and I know that my professor used this as his equation: ...
2
votes
3answers
817 views

Probability of having exactly 1 pair from drawing 5 cards

I have an exercise as follows: There is a collection of cards consisting of 52 cards (13 types and 4 colours each type). We draw 5 cards from the collection. Then what is the probability of having ...
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0answers
8 views

looking for asymmetric probability distribution

In physics, quantities are sometimes measured with both an upper and lower error. For example, I might say that an object's mass is $m = 10^{+0.1}_{-0.2} \text{ kg}$. In my case, these arise from ...
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2answers
23 views

Preliminaries: Combinatorial Analysis [on hold]

There are 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? What if 2 of the men are feuding and refuse to be on the committee together?
3
votes
1answer
545 views

Classic $2n$ people around a table problem

A total of $2n$ people, consisting of $n$ married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let $C_i$ denote the event that the members of couple $i$ ...
3
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5answers
93 views

$\sum\limits_{n=1}^\infty n(\frac{1}{2})^{n}$ [duplicate]

I am trying to find the expected value of the number of even numbers rolled before the first odd number when rolling a fair die until an odd number comes up. I arrived at $\sum\limits_{n=1}^\infty ...
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0answers
18 views

How to find the discrete probability vector given a transition probability matrix?

I have a transition probability matrix for the above Discrete Time Markov Chain I want to find the 'discrete probability vector' of this state space. My understanding is that the discrete ...
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votes
2answers
212 views

How minimize $\sum p_b \ln{p_b}$?

I have a multiset $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of integers in $A$. We ...
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votes
2answers
65 views

Discrete math: probability of picking certain hands with a preset condition

In 5-card draw poker, a player receives an initial hand of 5 cards, and is then allowed to replace up to three of her cards with the remaining cards in the deck. (b) Suppose that, among the initial 5 ...
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votes
2answers
25 views

MAP Estimator with Laplacian Noise

I need to calculate the MAP estimator of $ x $ in the following case: $$ \left [ \begin{matrix} {y}_{1}\\ {y}_{2} \end{matrix} \right ] = \left [ \begin{matrix} x\\ x \end{matrix} \right ] + ...
1
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2answers
48 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
2
votes
1answer
9 views

Bound on expectation of function of standard normal, $\mathbb{E}[\exp(Z^a)]$

I'm trying to find the maximum (or sup) of the value of $a$ such that $$\mathbb{E}[\exp(Z^a)]<+\infty$$ where $Z\sim \mathcal{N}(0,1)$. Obviously for $a=1$ the expectation is finite since it is the ...
0
votes
1answer
16 views

Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
5
votes
1answer
36 views

Probability of finding a Hamilton circuit in a graph

In short, I would like to know either/both the probability that there exists a Hamiltonian circuit within a graph, or the number of circuits expected to exist. (Without actually finding all the ...
0
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1answer
23 views

probability and applied statistics 3 [on hold]

given two urns, suppose urn 1 contains four black and seven white balls. urn 2 contains three black , one white , and four yellow balls . we select an urn at random and then draw a ball . what is the ...
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0answers
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Given two players competing, what is the probability of one getting ahead x wins vs the other getting ahead y

We have 2 players, A and B, competing. The probability that A wins a match is p, making the probability that B wins a match (1-p) = q. The game is won by player A as soon as he gets one more win than ...
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1answer
23 views

measurable function and composition of function

Show that if $f$ is a measurable function and $g$ is a continuous function on $\Bbb R$ then $g\circ f$ is measurable. please tell me how to prove it !
2
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1answer
33 views

what is the distributions of the random variable?

If moment generating function is $m(t)=[(1/3)e^{t}+(2/3)]^{5}$, then what is the distributions of the random variable?
0
votes
1answer
26 views

Conditioning twice?

I know that $P(X, Y)=P(X|Y)P(Y)$. How can we apply this to $P(X,Y|Z)$? We have already conditioned on $Z$, so can we condition it again on $Y$? Thanks!
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4answers
131 views

Probability Urn Question(Recurrence Model)

From an urn containing a white and b black balls, balls are drawn one by one at random according to the following rules: (i) at any drawing, if the ball drawn is white, then it is returned to the ...
0
votes
1answer
35 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...