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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5
votes
3answers
87 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
42 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
44 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 [closed]

[\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
19 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
49 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… [closed]

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
30 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
17 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 ...
1
vote
2answers
36 views

Probability of winning a 5 game series by winning 3 games where probability of winning each game is given

Two teams play a series of baseball games, team A and team B. The team that wins 3 of 5 games wins the series. The first game takes place in the stadium of the team A, the second in the stage of team ...
0
votes
1answer
32 views

How can I calculate the following probability?

Does it using poisson distribution to calculate the probability? Thank you!
0
votes
1answer
16 views
5
votes
1answer
44 views

For each pair of events A and B, find P(A|B) and P(B|A).

I've got a simple problem here, but I just want to ensure that I'm not losing a simple concept. Relevant equations: $$P(A|B) = \frac{P(A \cap B)}{P(B)},$$ $$P(A \cap B) = P(A) + P(B) - P(A \cup B)$$ ...
3
votes
2answers
36 views

Probability Independence - Determining if two sets are independent (drawing two cards)

I've got a few problems here that I feel pretty confident on. I am asking for confirmation on these answers. However, I am stuck on problem #3. Please let me know if you need more information. Two ...
0
votes
1answer
19 views

Probability of inequality of Random variables

Assume that $X_1,X_2,X_3$ are independent random variables with known distributions. How can I calculate the distribution of $P(X_1<X_2<X_3)$.
2
votes
5answers
118 views

How to prove $\frac{1}{1-p}=\sum_{n=r}^\infty {{n \choose r}p^{n-r}(1-p)^r }$

As we know $\frac{1}{(1-p)^{r+1}}=\sum_{k=0}^\infty{{k+r \choose k} p^k}$ and $\frac{1}{1-p}=\sum_{k=0}^\infty {p^k}$. But how to prove $$\frac{1}{1-p}=\sum_{n=r}^\infty {{n \choose r}p^{n-r}(1-p)^r ...
0
votes
1answer
14 views

Probability problem with random vectors

Problem Suppose that $10$% of the american population smokes dark cigarettes, $35$% smokes white cigarettes, $3$% smokes pipe and the rest of the population doesn't smoke. A group of $35$ persons was ...
0
votes
1answer
106 views

what is conditional distribution function of Y given N = n, correlation coefficient of Y and N, and what is effect of lambda on mean of Y

Y is a random variable defined as the sum of N independent Bernoulli trials where the probability of every bernoulli trial equalling '1' is equal to p. The number of Bernoulli trials N is itself a ...
1
vote
1answer
29 views

Rolling a pair of dice, conditional probability of neither die showing a 2 given they sum to 7.

This question is identical to this one, but I am not finding the explanation I am looking for in that question. My sample space would be $S = \{1,2,3,4,5,6\}^2$ and $P(s) = \frac{1}{36}$ for all $s ...
-2
votes
1answer
39 views

Expected value of $X^{2n}$ where $X \sim N(0,1)$ [closed]

The question is: Show that if $X ∼ N(0, 1)$ has the standard normal distribution then $E[X^{2n}] = \frac{2n!}{2^{n}n!}$ Hint: compute the integral $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} ...
0
votes
1answer
34 views

Random Variables $X$ and $Y$ have joint PDF, find marginal PDF

Random variables $X$ and $Y$ have joint PDF $$f(x,y) = \begin{cases} \frac{3}{8} x^2 e^{-2y}, & -2<x<2, \, y>0\\[2ex] 0,&\text{otherwise} \end{cases} $$ a) Find the marginal ...
1
vote
1answer
17 views

Given that you have exactly one pair, what is the probability you have two aces?

You are drawing 5 cards from a 52 card deck in a game of Poker. Here's what I've got so far, but I'm a bit stuck on how to proceed. Let $S = \{h \in 2^D : |h| = 5\}$ and $P(h) = \frac{1}{52 \choose ...
0
votes
1answer
47 views

Probability on roll of dice

This question gave me some pause: Suppose that you throw n dice. What is the probability that NO pair adds up to 8? I know there are C(n,2) possible pairs. And these are all the possible ways of a ...
0
votes
0answers
21 views

Two coins are tossed independently where P(head occurs when coin $i$ is tossed) = $p_i$, $i=1,2$.

Given that at least one head has occurred, the probability that the coin produce different outcome a) $\frac{2p_{1}p_{2}}{p_{1}+p_{2}-2p_{1}p_{2}}$ b) ...
0
votes
0answers
30 views

Dependence/Independence Problem in probability [closed]

Suppose a student is taught by $N$ teachers. The pdf of the marks that the student gets from $i$-th teacher is $p_{m_i}(x)$ and we assume that all $p_{m_i}(x)$'s are i.i.d i.e. ...
2
votes
1answer
17 views

Can you verify my solutions to these probabilities, given a coin is flipped 10 times?

I'm fresh into probability and I think it's important to ask a lot of questions since it seems probability really challenges your intuition. I'm working on the following problem, and have found a ...
1
vote
1answer
48 views

$P(X ≥ a) ≤ \frac{Var(X)}{Var(X) + a^2}$ when E(X) = 0

The full question is: Show that if $$E(X) = 0$$ then $$P(x\geq a)\leq \frac{Var[X]}{Var[X] + a^2}$$ Also show that there is an X for which the equality holds. I was able to note that: $$Var[X] = ...
0
votes
1answer
30 views

Probability of age of person renewing insurance?

Suppose that 5% of policy holders in a certain car insurance company do not renew their policies the following year. From the previous data 90% of people who renew policy are greater than or equal to ...
0
votes
1answer
16 views

conditional distribution determine joint distribution

$X,Y$ are two random variables and we know the distribution of $X|Y$ and $Y|X$, then I want a counterexample that $(X,Y)$ is not unique determined. this question is motivated by Can conditional ...
-6
votes
0answers
77 views

Bernoulli Process [closed]

Customers depart from a bookstore according to a Bernoulli process with parameter p = 0.15 (per minute). Each customer buys a book with probability 2/3, independent of everything else. Find the ...
0
votes
1answer
37 views

If I randomly select 6 books, what is the probability I…

I have 30 books. 5 are labeled classics, 10 are labeled mysteries, 7 are labeled science, and the rest are sports. If I randomly select 6 books, what is the probability I a) select at least 2 ...
1
vote
2answers
24 views

Let $X \sim \operatorname{Exp}(\lambda).$ Find $E[X^n]$, for all $n \in N$.

Let $X \sim \operatorname{Exp}(\lambda).$ Find $E[X^n]$, for all $n \in N$. I'm having difficulty figuring out how to do this problem and not sure quite how to solve this. I know that I'm supposed ...
3
votes
1answer
41 views

Probability of game ending with 10 quarters and 5 dimes?

This is a problem from a practice exam I was given, and I was wondering if you could help me figure it: A box has 5 quarters and 3 dimes. You draw a coin from the box and return it to the box with ...
-2
votes
1answer
28 views

asymmetric coin flip [closed]

So I know there are some questions here about coin fliping and probabilites, but I can't figure out my answer. So let's say we have 50 people that need to win a coin so they decide to flip it. The ...
1
vote
0answers
47 views

Condicional Expectation when $\mathbb{E}[X] = \infty$.

Let $(\Omega, \textit{F}_0, \mathbb{P})$ and $\textit{F} \subset \textit{F}_0$. Suppose $X \geq 0$ and $\mathbb{E}[X] = \infty$. Then there is a unique $Y \textit{F}$-measurable with $0 \leq Y \leq ...
1
vote
2answers
47 views

Dice Game with 1 die and Payoff Function

Imagine a dice game where you may repeatedly roll a die until you either decide to stop, or roll a 1, with the following payoff function (where k is the number on the die), $f(k) = 0$ when $k=1$ ...