1
vote
1answer
20 views

Distinct elements in the Union and Intersection of A and B

Take a set $x$ with $10$ distinct elements. Rule: Everytime you have two subsets, $A$ and $B,$ you also have $A\cup B$ and $A \cap B.$ What is the maximum number of subsets you can have such ...
0
votes
1answer
42 views

How to prove the binomial theorem

I'm trying to prove the following: $$ (p + q)^n = \sum_{x=1}^n \frac{n!}{x!(n-x)!}p^xq^{n-x}. $$ But I'm not sure where to start, would expanding the left hand side get me anywhere? Any tips or ...
0
votes
4answers
69 views

Chance of playing a game

You are offered a chance to play a game. the rules are simple. there are $100$ cards face down. Of these, $55$ say win and $45$ say lose. You begin with $10000$ dollars. You must bet $1/2$ of your ...
-2
votes
1answer
43 views

Set Theory Proof [closed]

How can I prove that $A \cup B = A \cup (A'B)$ It is a useful identity when proving : $ P(A \cup B) = P(A)+P(B)-P(AB)$ But I donĀ“t know where that identity comes from. Is there an analytical proof ...
0
votes
1answer
185 views

If $X' \leq X$ almost surely, is it possible to prove that $P(X = s) \geq P(X' = s)$?

With respect to my previous question, let us define $X$ as: $$ X = \sum_j^r l^j Y^j, $$ where $l^j \geq 0$ and $Y^j$, $j = 1, \ldots, r$ is a Bernoulli random variable which takes on values in ...
-1
votes
1answer
72 views

Mean and Standard Deviation self thought problem

I am 13 years old trying to teach myself about standard deviation and was wondering how this problem would look like. I know I am young to be learning this but I was reading about this and got ...
0
votes
2answers
106 views

How can one pass from “almost surely” to “surely”?

Several results (e.g in probability theory or using prob. theory) are stated in an almost surely phrasing (meaning the set of outcomes where this is not so has measure zero) How can one pass from ...
0
votes
1answer
32 views

Logistic Regression derivation

From the Wikipedia article http://en.wikipedia.org/wiki/Multinomial_logistic_regression: $ln \frac{\Pr(Y_i=1)}{\Pr(Y_i=K)} = \beta_1 \cdot \mathbf{X}_i $ $ln \frac{\Pr(Y_i=2)}{\Pr(Y_i=K)} = ...
0
votes
0answers
48 views

series-parallel system reliability equation proof

I am trying to device a proof for the series-parallel system reliability equation. The mathematical form of the equation is as follows: $Pr(\bigcap_{i=1}^{N}\bigcup_{j=1}^{M} A_{ij}(t)) = ...
0
votes
2answers
32 views

How to prove something at Normal distribution

$X\sim N(\mu,\sigma^2)$. $A,B$ are constants and $A\ne0$. How to prove that $AX+B\sim N(A\mu+B,A^2\sigma^2)$ ? Thank you!
0
votes
3answers
28 views

Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
2
votes
1answer
49 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
2
votes
2answers
41 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
6
votes
3answers
176 views

Is there a simple way to illustrate that Fermat's Last Theorem is plausible?

A first step in proving a theorem is true could be to show that it is plausible, so at least you then would have a general idea that it could be true and have something to start with in proving it. ...
2
votes
0answers
130 views

What is the optimal strategy for this 2 player game?

Let some finite array of integers is given initially. There is a number k which is initially '0'. In a move, a player will select a number from the array arr[i] and change k to gcd(k,arr[i]). Also, ...
0
votes
1answer
52 views

How to prove this simple fact from probability: $P(A)+P(B) = P(A\cup B)+P(A\cap B)$

I want to show that for events $A,B \subseteq \Omega$, $P(A)+P(B) = P(A\cup B)+P(A\cap B)$. This is obviously true, but I'm having trouble thinking of the way to prove it. Does anyone have a neat, ...
0
votes
1answer
22 views

How to prove that if $X\sim P(\lambda) \Rightarrow Var(X)=\lambda$?

How to prove that if $X\sim P(\lambda) \Rightarrow Var(X)=\lambda$? $P(X)$ means: Poisson distribution. Thank you!
0
votes
1answer
30 views

Expected value for the probability function $p * (1 - p)^{n - 1}$

$p * (1 - p)^{n - 1}$ is the function I came up with for this problem: ...
0
votes
1answer
48 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
0
votes
1answer
19 views

Prove the probability of n events intersecting

I'm trying to write a proof for this, but I don't know how to get started. Would proof by induction be the easiest way? If you could break it down into general steps I could wrap my head around, I ...
0
votes
0answers
50 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
1
vote
1answer
26 views

basic conditional probability proof

I having trouble with the following proof: $$P((A \cap B) \mid B) = P(A\mid B).$$ I get that $P(A\mid B) = P(A \cap B) / P (B)$, but I am unsure of how to proceed.
1
vote
2answers
50 views

$K$ events that are $(K-1)$-wise Independent but not Mutually/Fully Independent

I had the following question: Construct a probability space $(\Omega,P)$ and $k$ events, each with probability $\frac12$, that are $(k-1)$-wise, but not fully independent. Make the sample space as ...
1
vote
2answers
105 views

Using L'hopital's rule to solve problem.

Show that $$\lim_{x \to 0} \frac{-3x }{e^{x/3}}=0 $$ by L'hopital's rule. I know how to solve this without using L'hopital's rule. I was just reading about this and was wondering can we solve it ...
0
votes
1answer
40 views

Calculating Variance and Standard Deviation with probability distribution

The age [in years] $X$ of sewing machines to be reconditioned is a random variable with the following probability distribution: $f(x)=(1/972)x(18-x)$ for $0<x<18,$ and $f(x)=0,$ elsewhere. The ...
1
vote
0answers
45 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
0
votes
2answers
18 views

$A^c$ and $B^c$ are independent

I am trying to prove that, $A^c$ and $B^c$ are independent. My approach: $P(A^c \cap B^c)=P(A \cap B) - P(A \cap B)=P(A \cap B) \times (1-P(A \cap B) = P(A)P(B) \times ...
1
vote
2answers
32 views

Probability of winning a head on a coin

The problem I am asking is generated from this problem: Carla and Dave each toss a coin twice. The one who tosses the greater number of heads wins a prize. Suppose that Dave has a fair coin ...
10
votes
3answers
691 views

A very challenging probability question

In a certain 2-player game, the winner is determined by rolling a single 6-sided die in turn, until a 6 is shown, at which point the game ends immediately. Now, suppose that k dice are now rolled ...
0
votes
0answers
30 views

Subadditivity of events

I do not get this exercise: Let $A_i \in \mathbb{A}$ be a sequence of events. Show that: $P(\cup^{n}_{i=1} A_i) \leq \sum^n_{i=1} P(A_i)$ The solution is: Set $B_1 = A_1$, $B_i = A_i \setminus \ ( ...
1
vote
0answers
26 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
1
vote
0answers
31 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
2
votes
3answers
59 views

Proof expectation of bernoulli distribution

Suppose we have: $P(X=k) = (1-p)^k p$ $$E(X) = \sum^{\infty}_{k=0} kP(X=k)= \sum^{\infty}_{k=0} kp(1-p)^k = p(1-p) \frac{1}{p^2}=\frac{1-p}{p}$$ What I do not get is the step in the equation ...
2
votes
1answer
35 views

Is it possible to infer this relation without calculation?

Suppose $A\sim \mathscr{E}(\alpha)$ and $B\sim\mathscr{E}(\beta)$. Is it possible to argue that: $$\beta\,\mathbb{P}(A>B)=\alpha\,\mathbb{P}(B>A)$$ without calculating $\mathbb{P}(A>B)$ or ...
1
vote
1answer
55 views

Interviews Using Total Probability Theorem

Of all the job applicants at the Apex Company who come for a job interview ,$30$% are given the interview right away. The other $70$% are asked to wait in a waiting room. About $40$% of the time, ...
2
votes
1answer
62 views

Show that $\frac{1}{n}X_n\to 0$ a.s.

Show that for any sequence $(X_n)_{n\in\mathbb{N}}\in (L_{\mathbb{P}}^2)^{\mathbb{N}}$ of identically distributed random variables it is $\frac{1}{n}X_n\to 0\text{ a.s.}$. The professor ...
0
votes
0answers
16 views

Prove that the following function of binary random variables is monotonic

Consider a binary random variable $y$ over the space $\mathcal{Y} = \{+1, -1\}$ such that $\Pr(y = 1) = q$. Consider also $r$ binary random variables $y^1, \ldots, y^1$ over the space $\mathcal{Y}$ ...
0
votes
1answer
59 views

Probability Theory proof question

Problem: In football, a coin known to be unfair is tossed to see who receives the first kickoff. Your team has a peculiar curse in that the probability of winning the game given that they won ...
4
votes
0answers
163 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
2
votes
0answers
71 views

A follow-up to the regular hexagon question

This is a follow-up to the regular hexagon question. The problem statement was: Suppose we have a sphere and more than a half of its surface is red. Prove or disprove that we can place all ...
3
votes
1answer
216 views

Proof on Brownian Bridge

PROBLEM Let $U_t$ be a Brownian bridge on $[0,1]$ and let $Z$ be a standard normal random variable independent of $U_t$. $(a)$ Prove that the process $W_t = U_t + tZ$ is a brownian motion. $(b)$ ...
1
vote
1answer
26 views

Joint distribution proof

I am trying to study for an exam and I am kind of lost on how my professor came to a particular result on his practice exam. Let $W$ be an exponentially distributed random variable with $\lambda = 2$ ...
2
votes
1answer
86 views

Difficult Discrete/Probability Problem

Here's the question: For a function $f:[n]\rightarrow[n]$, where $n$ is the set $\{1,2,3,\ldots,n\}$, define the inverse complexity, $ic(f)$ as the number of ordered pairs $\langle i,j \rangle$ ...
2
votes
1answer
45 views

Variance of the Random Variable $|im(f)|$ where $f:[n] \rightarrow [n]$

Here's a question: Let $f$ be picked randomly from the set of all functions from $[n]$ to $[n]$, where $[n]$ is the set $\{1,2,3,\ldots,n\}$. Give a closed-form expression for the variance of the ...
0
votes
0answers
116 views

Probability Proof by Induction

How can I prove with induction that if one of two events can occur on any given day, $A$ and $B$, and given that if one occurs on one day, the probability that it occurs again on the next day is $1-p$ ...
1
vote
0answers
23 views

Chi-squared test property

I have got the following: $$\sum_{i=1}^{k-1} \frac{O_i - N p_i }{N p_i + (1-p_i)} \sim \chi^2_{k-1} $$ How to prove that $$\sum_{i=1}^{k} \frac{O_i - N p_i }{N p_i} \sim \chi^2_{k-1} $$? Where: ...
0
votes
1answer
98 views

Max function with probabilities

I have the following: $$p(Y<y) = p(\max(x_1, x_2, \ldots, x_t) < y)$$ Where $x_1, x_2, \ldots, x_t$ are independent(they come from a sample) why the following is true? $$p(\max(x_1, x_2, ...
2
votes
1answer
99 views

Equality of sets when minimizing Shannon's Entropy

Let $P = \{p_1, \ldots, p_n\}$ be a set of probabilities, i.e., $0 \leq p_i \leq 1$. $P$ is such that $\sum_{p_i \in P} p_i = 1$. I have a set of actions $\mathcal{A} = \{a_1, \ldots, a_N\}$ that can ...
0
votes
4answers
78 views

Elementary set proof

On a statistics trial exam I encountered the following proof I was supposed to give but I have no idea how to start with this proof and solve it: $P(A\cap B)$ $\geq$ $1 - P(A') - P(B')$ where $A'$ is ...
2
votes
1answer
115 views

Where is the fallacy in this coupling argument of two Bernoulli variables?

With respect to the scenario introduced in Prove the monotonicity of the expectation of a binary random variable function, let us now suppose that the function: $$\begin{align*} f(\mathcal{J}) = ...