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 ...

learn more… | top users | synonyms (2)

1
vote
0answers
8 views

Mean Squared Error for a Quantized Estimator

Let $P$ be a distribution supported on the unit interval. Suppose we are interested in estimating the mean value (denoted $\theta$) of $P$ based on data. Then if $X_1,\ldots,X_n \in [0,1]$ is data ...
-1
votes
1answer
30 views

Probability question based on Bayes theorem [closed]

Professsor Rosencrantz flips a fair coin. Professor Guildenstern flips a fair coin twice. What is a probability that professor Rosencrantz obtains more heads than professor guildenstren?
0
votes
0answers
19 views

The limit of probability mass functions is also a mass function

Suppose $\{X_n\}$ are discrete random variables with respective mass function $\{p_n\}$. Prove that if there exist a function $p$ such that $\lim_{n\rightarrow \infty}\sum_{z\in R}|p_{n}(z)-p(z)|=0$, ...
1
vote
1answer
36 views

Find probablity that player A go bankrupt (of Markov chain)

PLayers $A$ and B play in following game: With prob=$p$ player $A$ loses one coin, then player $B$ get one coin. With prob = $q$ player $B$ loses one coin and player $A$ get one coin. At begin they ...
1
vote
3answers
33 views

Getting the standard deviation from the pdf

A normally distributed random variable with mean $\mu$ has a probability density function given by $\dfrac{\gamma}{\sqrt{2\pi\sigma}}$ $\exp(-\dfrac{\gamma ^2}{\sigma} \dfrac{(x-\mu)^2}{2}) $ So ...
1
vote
3answers
34 views

Expected value of $X^2$

Let $X$ be a normally distributed random variable with $\mu = 4$ and $\sigma = 2$. If $E[X]$ denotes the expectation of $X$, then what is the value of $E[X^2]$? So I don't know exactly how to ...
0
votes
2answers
28 views

Minimizing Expected Value

I have a problem which wants the c value that minimizes E[(X-c)2] I started with E[(X-c)2] = E[X]2 -2cE[X] + c2 but couldn't continue on this. Could you please help me on finishing this problem. ...
0
votes
0answers
29 views

Is it possible to compute a probability between two vectors? [closed]

Is it possible to compute the probability between two vectors of the same length? I understand there are plenty of divergence and distance metrics for computing vector pairs but I'm looking for ...
0
votes
1answer
29 views

Help me get started on calculating CDFs/PDFs

Reading in a probability book. Lots of formulas, but almost no examples, and no classes to attend, so I'd like some help with figuring out CDFs and PDFs for transformations of two random variables. ...
0
votes
2answers
41 views

Independent and identically distributed random variables

Let $Y=1/4(X_1 + X_2 + X_3 + X_4)$, where $X_1$, $X_2$, $X_3$ and $X_4$ are i.i.d. r.v.s (independent and identically distributed random variables) with a Cauchy pdf $$f_X(x) = \frac{a}{\pi(x^2 + ...
2
votes
1answer
46 views

Statistically Independent Random Variables

Problem: For the statistically independent ramdon variables X and Y with fX(x)=1, 1≤x≤2, and fY(y)=e-(y-1), 1≤y<∞, determine fZ(z) where Z=X+Y I couldn't find a ...
1
vote
1answer
34 views

Is a density function integrable?

Let $X \in \mathbb{R}$ be a continuous random variable with density function $f$ (i.e. $f(x)\geq 0$ and $\int f(x)dx=1$). Does this mean that $\int |f(x)|dx < \infty$ i.e. $f \in L^1$? (The reason ...
0
votes
1answer
46 views

Expected value of the product of i.i.d random variables

Assume we have random variables $$X_i \,\,\,\ \text{ i.i.d } \,\,\ i\in[1:n]$$ with expected value $$\mathbb{E}[X_i] = \frac{1}{2}$$ Now let us compute the following expected value of the product of ...
0
votes
2answers
35 views

Help me prove this an absoulte inequality [closed]

Let $0\leq\pi_{i}\leq1\quad s.t. \sum\pi_{i}=1$, and $f(x;\phi_{i})$ be a normal pdf paired with $\pi_{i}$ Then, I need to prove below $\sum\pi_{i}f(x;\phi_{i})^2\geq(\sum\pi_{i}f(x;\phi_{i}))^2$ ...
2
votes
2answers
61 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of H’s and T’s. Let N denote the number of tosses until you see “TH” for the first time. For example, for the sequence HTTTTHHTHT, we needed N = 6 ...
0
votes
2answers
25 views

Probability of $A \cup B \cup C$

the question is this: a random man draws $1$ card from a deck. let $A$=this card is spade $B$= this card is red $C$= this card is a picture card(so jack,Q,and K) I need to find $P(A \cup B \cup C)$ ...
3
votes
1answer
65 views

Probability question about whether the landing spot of a tossed coin covers a specified point in space

I am having some trouble with the following question from Chapter 5 review of Pitman's book titled Probability: A coin of diameter 1 inch is tossed in the air and caught in an empty soup can of ...
0
votes
1answer
20 views

The Probability of 4 heads given the first toss is a head

The Question Alice tosses a fair coin seven times. Find the probability that she tosses 4 heads given her first toss is a head. Then, find the probability that she tosses 4 heads given her first and ...
3
votes
2answers
101 views

Expected number of days for books to return to their original position

An ordered vertical stack of n books is on my desk. Every day, I pick one book uniformly at random from the stack and put the book on the top of the stack. What is the expected number of days before ...
2
votes
1answer
37 views

Tetra Master Dice Rolls

In the game Tetra Master, two players play a card with a number n between 0 and 15, inclusive. Then, both players roll a 16-sided die numbered from 0 to 15. Both players than add up their dice rolls ...
-1
votes
2answers
42 views

Does P(A|B) = P(A) imply P(B|A) = P(B)? [closed]

If P(A|B) = P(A), does that necessarily mean that P(B|A) = P(B)?
4
votes
6answers
2k views

A die is rolled 3 times. What is the probability that a five is rolled at least twice?

The probability of not getting a five is $(\frac56)^3$, and I figure the probability of getting at least one 5 is $1-(\frac56)^3$, but I don't know how to figure out if it is rolled at least twice. ...
3
votes
2answers
29 views

Mathematical justification for incorporating a conditional event in expectation?

Let $X_1,X_2,\dots$ be independent and identically distributed random variables. Furthermore, consider the sum $$ Y = X_1 + X_2 + \dots + X_N $$ where the number of terms $N$ is itself a random ...
0
votes
3answers
35 views

Finding value of $p(x)$ given an MGF

Question: The MGF of the independent discrete random variable $X$ is given by $$M_X(t) = \left(\frac{1}{2}e^{2t} + \frac{1}{2}e^{4t}\right)^7$$ Find $p_X(15)$ I have been staring at this ...
0
votes
2answers
29 views

Unlikely Events - Where does the Natural Force of Mean Reversion Kick In?

If you play a 100x game of coin flip using a fair coin with a friend. You decide to opt out from picking H-or-T on any of the games until the 51st flip but you get to see the outcome of the flips ...
-5
votes
1answer
19 views

What is the probability of rolling at least one 3 in 3 rolls? [closed]

-I am a little confused -please show full solution -I have to present the solution in front of my class tomorrow Thanks :)
2
votes
1answer
50 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
3
votes
1answer
32 views

Does wikipedia state the definition of probability correctly?

In the wikipedia article on probability http://en.wikipedia.org/wiki/Probability it says: To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a ...
4
votes
0answers
121 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
0
votes
1answer
19 views

Probablity - Calculate $E\left[X^2\right]$ using MGF

i know that $X\:\:U\left(\:r\:\:,\:-r\right)\:\:\:\:0<r<\sqrt{3}$. So, want to Calculate $E\left[X^2\right]$ using MGF. I know that MGF of discrete uniform random variable is ...
0
votes
2answers
34 views

Probability using combinatorics problem

The problem is simple: "Find the probability of getting no aces with four dice". Now, i'm supposed to solve this using combinatorics. So, I see two ways of doing this. First: considering my sample ...
1
vote
0answers
17 views

Probability question about sums of random variables and conditional expectation

Let $X_1, X_2,...$ be independent and identically distributed random variables with distribution $P(X_i = x) = p$ if $x=1$ and $P(X_i = x) = 1-p$ if $x=0$. Let N be a Poisson random variable with ...
2
votes
2answers
19 views

How many numbers are in a numbering system with the basis 15 and 4 digits, where the digit sum equals 15

First of all, my question is very similiar to this one: How many numbers between $100$ and $900$ have sum of their digits equal to $15$? but i didn't quite understand how to adapt it to my problem, so ...
-1
votes
0answers
14 views

Epsilon Differential privacy using laplace distribution

Given Database X={x_1 ,x_2 ,...X_n} where x_i represent a bit Let f(x) = Sum x_i for 1 <= i <= n Let Y ~laplace(1/(epsilon)) I succeeded proving that Mechanisms: M1(x)= Y-f(x) and M2(x)= ...
-4
votes
1answer
53 views

Precise mathematical definition of Probability.

Is there a precise definition of probability in mathematical terms? For example we have a precise definition of limit in terms of epsilon and delta,so what is the mathematical definition of ...
-1
votes
1answer
24 views

if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$?

In handouts provided by a professor I read: if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$. It does not feel right to me. $X ...
0
votes
2answers
25 views

Creating a probability density function from a word problem

I am taking a course related to probabilities and as a primer we are given some word problems, I have somehow slipped by in my earlier classes and never have taken a classes on such subjects. I was ...
-1
votes
0answers
13 views

Odds of wining sweepstakes vs price of ticket

I've seen lots of similar questions but none answer my question. There is an online raffle (so no bend ticket answers) with $2000$ tickets total. Each ticket costs $5$ points, $1$ prize of $5000$ ...
0
votes
1answer
10 views

Probability Tree Diagram artist 60% win 40% lose [closed]

An artist wins a prize 60% of time when she enters an art show. Draw a tree diagram and use it to determine the probability that she will win exactly once out of two art shows.
2
votes
0answers
40 views

$e^{-d|z|^\alpha}$, $d\geq0,0<\alpha\leq2$, is characteristic function of a stable distribution

Problem: Prove that $e^{-d|z|^\alpha}$ is characteristic function of a stable distribution, if $d\geq0$ and $0<\alpha\leq2$. A note on the definition of stable: Note that a measure $\mu$ ...
-2
votes
0answers
23 views

Probability (Please see picture below) [closed]

On a scratch card you win if you find a sun in the first square you scratch off. Here are the scratch cards before the suns are covered. $$\mathbf{A}\ \ \ \ \ \ \ ...
2
votes
1answer
15 views

Clarifying the elementary calculus used in this statistics problem

Let $X \sim N(\mu, \sigma^{2})$ and $Y = \alpha X + \beta$ for $\alpha > 0$. I'm looking at a demonstration that $Y = \alpha X + \beta \sim N(\alpha\mu + \beta, (\alpha\sigma)^{2})$, and find ...
1
vote
0answers
12 views

Counting distinct positive valued k-tuples that sum to n where each entry can be no greater than some value.

This is motivated by the desire to count the number of ways two dice can form the sums 2,3,4,...,12 respectively. We can safely use the stars and bars method for 2,3,4,...,7 where the number of ways ...
2
votes
1answer
63 views

improbable sum of random variables

Let $U$ be a uniform random variable on the interval $[0,1]$. It is exceedingly unlikely that $U$ can be written as a sum $U = X + Y$ where $X$ and $Y$ are independent identically distributed random ...
1
vote
0answers
40 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
0
votes
0answers
19 views

Find parameters that events are independent

On three walls of the cube with four of walls are placed (for each wall one) dots: green ($p_1$), red($p_2$), blue($p_3$). On the fourth wall are three dots, green, red, blue. ($p_4$) What value of ...
0
votes
0answers
8 views

Conditional expectation binomial

Toss $n$ coins. What is the conditional expectation of the number of heads given the number of heads among the first x tosses? I let $X \sim Bin (x, 0.5)$ and $Y \sim Bin (n-x , 0.5)$ where $X$ is ...
0
votes
2answers
24 views

What is right solution for this probability problem?

This drug can cure $90$% of all diseases. What is probabilty of successful healing at least $18$ people of $20$ people, who have taken the drug? What is the right solution and why? From my point of ...
0
votes
3answers
28 views

Is this a conditional probability or not?

Suppose that the telephone calls during one minute time follow a Poisson distribution with mean=4. If people can handle at most 6 calls per minute, what is the probability that the people will receive ...
0
votes
1answer
38 views

Generating a data set with given mean and variance

Suppose we have to create n integers in a given range say between 1 and 1000 with given mean and variance .My question is :Is there an algorythm that can tell us whether such a data set exists and ...