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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2
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1answer
38 views

If you choose 4 of these lamps at random, what is the probability that none need to be replaced during the first $150$ hours of use?

The lifetime, in hours, of each lamp produced by a certain company, is a random variable with density function given by $$f(t)=\begin{cases}100/t^2,& t>100\\ ...
3
votes
0answers
38 views

Probability problem of fishes in a lake

Exercise In order to estimate the number $N$ of fishes in a lake, a fisherman executes the following procedure: in the first step, he captures $n$ fishes and after marking them, he returns them to ...
-14
votes
0answers
66 views

Terrorist attack in New York [on hold]

What's the probability of another attack in New York? Can we compute that? Is it zero? The last attack in New York was on sept 11th 2001. There was also an attack on Boston. Is the system of ...
-3
votes
1answer
35 views

Use a $Z$ table to find $P(-1 < Z < 1)$. [on hold]

Can someone help with these problems, please? Use Appendix Table III to determine the following probabilities for the standard normal random variable $Z$: (a) $P(-1 < Z < 1)$ (b) $P(-2 < Z ...
0
votes
1answer
27 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
2
votes
1answer
25 views

probability problem with Poisson distribution

Problem A retailer knows that the demand of boxes is a random variable with Poisson distribution of parameter $\lambda=2$ boxes per week. The retailer completes his stock on monday so as to have four ...
1
vote
1answer
26 views

Type 2 Error Question - How to calculate for a two tailed?

The modulus of rupture (MOR) for a particular grade of pencil lead is known to have a standard deviation of 250 psi. Process standards call for a target value of 6500 psi for the true mean MOR. For ...
1
vote
2answers
59 views

Bayesian urn questions

There are two urns, each with four ping-pong balls. In one urn, three of the balls are red, and one is white; in the other, three are white, and one is red. Without knowing which urn you are choosing, ...
4
votes
3answers
84 views

Winning All Levels in a Game

There are $L$ levels in a game. In each turn of the game, you go through each level one by one and try to complete it. The goal is to complete all levels of the game. The probability of completing any ...
-4
votes
0answers
20 views

Determine p value of the statistics [on hold]

Given X is a geometric random variable with pmf and cdf $$p(x) = 0.153(1-0.153)^{x-1} $$ and $$F(x) = 1-(1-0.153)^x$$ A statistics S is defined by $S = \min\{X_1, X_2,X_3,\ldots,X_n\}$ If the ...
0
votes
0answers
39 views

Facebook Data Science Question (Expected Payout and Probability)

I saw this question on Glassdoor and couldn't seem to find a answer to validate mine anywhere: You're at a casino with two dice, if you roll a 5 you win, and get paid $10. What is your expected ...
0
votes
1answer
35 views

How to prove MLE of theta is unbiased?

Let $X_1, X_2, . . . , X_n$ be a random sample from a uniform distribution on $[0, \theta]$. Suppose results $x_1, x_2, . . . , x_n$ are observed. Since $f(x) = 1/\theta$ for $0 \leq x \leq \theta$, ...
1
vote
2answers
56 views

Expected number of coin tosses

A fair coin is tossed until either 4 heads or 9 tails obtained ( total). What is the expected number of tosses? Edit: I calculated the probability. It is 10% for 4 heads and 12% for 9 tails. But how ...
0
votes
1answer
33 views

Uniform Distribution Question - Help Needed

Let $X_1, X_2, . . . , X_n$ be a random sample from a uniform distribution on $[0, \theta]$. Suppose results $x_1, x_2, . . . , x_n$ are observed. Since $f(x) = 1/\theta$ for $0 \leq x \leq \theta$, ...
1
vote
2answers
31 views

Probability Question - Moment Generating Function

$$ f(x) = \begin{cases} xe^{-x}, & \text{x ≥ 0} \\ 0, & \text{elsewhere} \end{cases}$$ Q: Find the Moment Generating Function of X. Hi, I was trying to solve this question by putting the ...
2
votes
3answers
58 views

Picking two random points on a disk

I try to solve the following: Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those ...
-1
votes
0answers
28 views

how find an event with probability $6.6p^2q$ [on hold]

How to find an event with the following probabilities $6p^2q$,$6.6p^2q$,$6.75p^2q$,$3.9pq$,$4pq$ using independent bernoulli trials?
1
vote
0answers
20 views

Posterior of Normal with prior Cauchy

Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$ To find the credible set I need to find the distribution of $f(\theta\mid x)$, but ...
2
votes
0answers
27 views

Can someone help me balance this game (probability question) [on hold]

A team of 9 vs a team of 1. Each round each of "the 9" roll a die to "attack" and "the 1" rolls 9 dice to "defend", the nine dice are preassigned to attackers before the roll, "the 1" cannot choose ...
-2
votes
1answer
12 views

How do i find the Maximum Likelihood Function Estimator [on hold]

Im trying to find the estimator for $\mu$ but the lecture notes i have don't explain very well. The question is find the maximum likelihood function estimator of: $$L(\mu : x) = c. ...
-1
votes
2answers
38 views

Finding the PDF of Y=X-2 [on hold]

I am given the following PDF of random variable $X$: $$f(x)= \begin{cases} e^x & \text{for }x<0, \\ 0 & \text{otherwise}. \end{cases}$$ a) Compute $E(e^x)$: Here is my work: ...
0
votes
0answers
17 views

Change of variable formula for density function

We all are aware of the change of variable formula whereby if $$[A, B] = g(X, Y) $$ and g is invertible, then the joint density function of A, B is given by $$f_{ab} (A, B) =1/|J| f_{XY} (g^{-1}(a, ...
1
vote
0answers
44 views

Verify I proved that $P(A) = P(A\mid B)P(B) + P(A\mid\bar B)P(\bar B)$ correctly?

Proof. Let $A$ and $B$ be events in a sample space $(S,P)$. Suppose $0 \le P(B) \lt 1$. $P(A) = P(A\mid B)P(B) + P(A\mid\bar B) P(\bar B).$ $P(A) = P(A \cap B) + P(A \cap \bar B)$ by ...
1
vote
2answers
30 views

Positive Expectancy with Random Chance

A conversation came up at work about positive expectancy. I am having difficulty getting the same answer as the guys so I am throwing the question out to you folks... Any help is appreciated. A game ...
1
vote
1answer
25 views

Finding distribution of random variable

During my exam there was the following question which I could not answer: Let $X_1, X_2$ be real valued random variables. Assume that $X_1$ is exponentially distributed. Given that ...
0
votes
1answer
40 views

How would you simulate Brownian motion with a die?

You can simulate Brownian motion on $[0, 1]$ for instance by splitting it into $K$ intervals and at each time $k \Delta t$ add $N(0, \Delta t)$ to your running total, where $\Delta t = 1/K$. If you ...
0
votes
0answers
22 views

show that $U_n$ converges to $0$ in $L^1$ and almost surely.

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
5
votes
3answers
86 views

Probability with n dice

I'm studying probability and am currently stuck on this question: Let's say we have n distinct dice, each of which is fair and 6-sided. If all of these dice are rolled, what is the probability that ...
1
vote
1answer
41 views

Expected value of exponential function

Suppose two identical component are connected in a piece of factory equipment. The two lifetimes X1 and X2 are independent each having exponential distribution with beta =2. The value of the equipment ...
0
votes
0answers
43 views

Sequence of non-independent coin tosses

Suppose that a sequence of coin tosses is due to be performed. Let $p_i$ denote the probability that the $i$th coin toss lands on Heads and let $X_i$ denote the corresponding indicator random variable ...
-3
votes
0answers
19 views

Function of two continuous random variables. find CDF [on hold]

[\begin{array}{l}{\rm{Let X be a continuous random variable with uniform distribution on }}\left[ {0,1} \right].{\rm{ }}\\{\rm{Let Y be a continuous random variable with uniform distribution on ...
0
votes
0answers
12 views

integration with delta function

Is there any way to calculate the following expression: $$\{\frac{\partial}{\partial t}\int|(1-t)p(x)+t\delta_{x_0}(x)-c|dx\}_{\text{at t=0}}.$$ Here, $p$ is a probability density function, ...
2
votes
0answers
18 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
0
votes
1answer
48 views

Average to collect baseball cards

A young baseball fan wants to collect a complete set of 262 baseball cards. The baseball cards are available in a completely random fashion, one per package of chewing gum. The fan buys two packets ...
1
vote
1answer
28 views

For two independent events $A$ and $B$, find $P(A \cap B^c|A \cup B) $

For two independent events $A$ and $B$, find $P(A \cap B^c|A \cup B)$. Futhermore, we know the probability $P(A)$=0.4 and $P(A\cup B)$=0.5. I thought that since $$P(A|B)= \frac{P(A\cap B)}{P(B)},$$ I ...
-1
votes
1answer
39 views

Let X be a continuous random variable with pdf… [on hold]

a.) Let X be a continuous random variable with pdf $f_x(t) = \exp[-t-e^{-t}]$ for all t in the reals. Find $F_X(x)$ My solution is $$F_X(x)= P(X \le x) = ...
1
vote
0answers
28 views

Finding the marginal density

The joint probability density function of $X$ and $Y$ is given by $$f(x, y) = 1/y^2 , 0< x< 1, y\geq 1 $$ $[I]$ - Find the joint density function of $U = XY$ and $V = X/Y$ $[II]$ - What are ...
0
votes
0answers
9 views

Closed-form expression for conditional expected value

Say we have $n$ applicants $a_0...a_n$ waiting to interview for a job, in order. Any number of them may be accepted, but with probability that decreases with the number of applicants already accepted ...
0
votes
2answers
29 views

Defective items in a bag

Can anyone help me in these questions: I am not sure if I am doing it in the right way. A bag contains 20 items with 5 defective items. Items are sampled at random one at a time. What is the ...
1
vote
1answer
17 views

Poisson distribution- mosquitos question

Can anyone help me in these questions? I am not sure if I am thinking in the right way When one is camping, mosquitoes are observed to land on one’s body at an average rate of 3 per minute. Using ...
2
votes
2answers
30 views

Identifying a distribution from its moments

I came across a random variable whose sequence of central standard moments empirically seems to be $0, 1/2, 0, 3/2, \dots$. (That's as far as I could compute.) Is this a well-known distribution?
0
votes
1answer
16 views

Show that $\sum\limits_{i=1}^{\inf} p_i \prod\limits_{j=1}^{i - 1}(1 - p_j) + \prod_{i=1}^{\inf}(1 - p_i) = 1 $

The question comes from Hoff's "A First Course in Probability" book. Let $p_i$, $i = 1, 2, ...$ be probabilities (so that $0 \leq p_i \leq 1$, and show the that the equation in the title holds, ...
0
votes
1answer
13 views

Continuous and discrete random variables defined on the same probability space?

I am confused on the definition of continuous/discrete random variables defined on the same probability space. Consider the random variables $X,Y$ defined on the same probability space $(\Omega, ...
0
votes
0answers
19 views

Estimate for average probability of Ito diffusion falls into an interval

Denote $E^x(X_t)$ be the solution to a Ito diffusion starting with $X_0=x$. Let $K\subset \mathbb{R}$ be a compact subset. I also assume $X^x_t$ has transition probability $p(t,y,x)$. Currently I am ...
2
votes
2answers
36 views

I cannot figure out part a) ii) and iii) in the following question

"Two students, Karl and Hanna, play a game in which they take it in turns to select a card, with replacement, from a well-shuffled pack of 52 playing cards. The first person to select an ace wins the ...
1
vote
5answers
56 views

A fair coin is continually flipped until heads appears for the 10th time. Find the number of expected tails

A fair coin is continually flipped until heads appears for the 10th time. Find the number of expected tails. Im very lost in this problem, can someone help? I think I have to use neg binomial, but ...
0
votes
1answer
16 views

For this probability question, should I consider him stepping back and then forward again?

"A delirious man stands on the edge of a cliff and takes random steps either towards or away from the cliff’s edge. The probability of him stepping away from the edge is $\frac{3}{5}$ , and towards ...
0
votes
2answers
30 views

Why this integral equals to $\Gamma(4)10^4$

I'm stuck with this equation: $$\int_0^{\infty}y^3 e^{-\frac{y}{10}}~~dy=\Gamma(4)10^4.$$ In this equation, $\Gamma$ stands for Gamma function. I don't know where does $10^4$ come from. Anyone can ...
0
votes
0answers
10 views

Connect the MGF of the Survivor, Cumulative and Mass disttributions

Assume that $X$ has a known distribution $P_X$, with a generating function $\hat P_X$. What relationship links $\hat P_X$ with the MGF of X's CDF ($\hat C_X$) and SDF ($\hat S_X$). Would that ...
0
votes
2answers
49 views

Prove $P(X=k \mid X+Y=n) = \frac1{n+1}$

Let $f_X(k) = f_Y(k)= p(1-p)^k~$ for all $k = 0,1,2,\ldots$ for some $0 < p < 1$. Show that for any $n \ge 0$ $$P(X=k \mid X+Y=n) = \frac1{n+1}$$ for any $0 \le k \le n$. What is confusing ...