Let $X\sim Geom(0.75)$. Find the probability that X is divisible by 3

Zack Chase
2022-09-17
Answered

Let $X\sim Geom(0.75)$. Find the probability that X is divisible by 3

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asked 2022-07-23

100 Gambles of Roulette

I am stuck on the following problem: You go to Las Vegas with $1000 and play roulette 100 times by betting $10 on red each time. Compute the probability of losing more than $100. Hint: Each bet you have chance 18/38 of winning $10 and chance 20/38 of losing $10.

So essentially this is recurring 100 times. Now each time you do it you have an 18/38 chance of winning and 20/38 chance of losing. You have a greater chance of losing than winning, so based on speculation you need to lose 56/100 in order to lose more than $100, to counterbalance your winnings . How would I set up an equation to solve for this?

I am stuck on the following problem: You go to Las Vegas with $1000 and play roulette 100 times by betting $10 on red each time. Compute the probability of losing more than $100. Hint: Each bet you have chance 18/38 of winning $10 and chance 20/38 of losing $10.

So essentially this is recurring 100 times. Now each time you do it you have an 18/38 chance of winning and 20/38 chance of losing. You have a greater chance of losing than winning, so based on speculation you need to lose 56/100 in order to lose more than $100, to counterbalance your winnings . How would I set up an equation to solve for this?

asked 2022-08-14

Geometric or binomial distribution?

A monkey is sitting at a simplified keyboard that only includes the keys "a", "b", and "c". The monkey presses the keys at random. Let X be the number of keys pressed until the money has passed all the different keys at least once. For example, if the monkey typed "accaacbcaaac.." then X would equal 7 whereas if the money typed "cbaccaabbcab.." then X would equal 3.

a.) What is the probability $X\ge 10$?

b.) Prove that for an random variable Z taking values in the range {1,2,3,...}, $E(Z)=$ Summation from $i=1$ to infinity of $P(Z\ge i)$.

c.) What's the expected value of X?

First, is this a binomial distribution or a geometric distribution? I believe it is a binomial but my other friends says that it is geometric. As for the questions above, for a can I just do $1-P(X=9)\text{}\text{or}\text{}1-P(X9)$, but I don't know how I will calculate $X<9$, I would know how to calculate $P(X=9)$, I don't know how to do b or c.

A monkey is sitting at a simplified keyboard that only includes the keys "a", "b", and "c". The monkey presses the keys at random. Let X be the number of keys pressed until the money has passed all the different keys at least once. For example, if the monkey typed "accaacbcaaac.." then X would equal 7 whereas if the money typed "cbaccaabbcab.." then X would equal 3.

a.) What is the probability $X\ge 10$?

b.) Prove that for an random variable Z taking values in the range {1,2,3,...}, $E(Z)=$ Summation from $i=1$ to infinity of $P(Z\ge i)$.

c.) What's the expected value of X?

First, is this a binomial distribution or a geometric distribution? I believe it is a binomial but my other friends says that it is geometric. As for the questions above, for a can I just do $1-P(X=9)\text{}\text{or}\text{}1-P(X9)$, but I don't know how I will calculate $X<9$, I would know how to calculate $P(X=9)$, I don't know how to do b or c.

asked 2022-08-19

Probability in higher dimension

There are n non negative numbers ${x}_{i}<1$. It is given that ${S}_{1}=\sum _{i=1}^{n}{x}_{i}=1$. What is the probability that

${S}_{2}=\sum _{i=1}^{n}{x}_{i}^{2}<c$ where

$\frac{1}{n}<c<1$

Assume that each ${x}_{i}$ is uniformly distributed.

Now I tried looking at this problem in terms of integrals:

$P(squares|sum)=\frac{P(squares\cap sum)}{P(sum)}=\frac{something}{{\int}_{0}^{1}{\int}_{0}^{1-{x}_{n}}{\int}_{0}^{1-{x}_{3}-{x}_{4}-...-{x}_{n}}...{\int}_{0}^{1-{x}_{2}-{x}_{3}-...-{x}_{n}}d{x}_{1}d{x}_{2}...d{x}_{n-1}d{x}_{n}}$

Now for the numerator, if we substitute the constraint in the inequality, we get

${S}_{2}+(1-{S}_{1}{)}^{2}=c$

${S}_{2}+(1-{S}_{1}{)}^{2}=c$

Now how to get the numerator integral, I'm not quite sure.

Also, is there any better, more elegant way of doing this problem? (I thought of using vectors, where the problem can be written as: $\overrightarrow{x}.\overrightarrow{1}=1$ then $P(||\overrightarrow{x}||<\sqrt{c})=?$. But then we would have to model the distribution of $||\overrightarrow{x}||$ which I have no idea about)

Or maybe a more direct approach? Like the ratio of areas (or volumes?).

There are n non negative numbers ${x}_{i}<1$. It is given that ${S}_{1}=\sum _{i=1}^{n}{x}_{i}=1$. What is the probability that

${S}_{2}=\sum _{i=1}^{n}{x}_{i}^{2}<c$ where

$\frac{1}{n}<c<1$

Assume that each ${x}_{i}$ is uniformly distributed.

Now I tried looking at this problem in terms of integrals:

$P(squares|sum)=\frac{P(squares\cap sum)}{P(sum)}=\frac{something}{{\int}_{0}^{1}{\int}_{0}^{1-{x}_{n}}{\int}_{0}^{1-{x}_{3}-{x}_{4}-...-{x}_{n}}...{\int}_{0}^{1-{x}_{2}-{x}_{3}-...-{x}_{n}}d{x}_{1}d{x}_{2}...d{x}_{n-1}d{x}_{n}}$

Now for the numerator, if we substitute the constraint in the inequality, we get

${S}_{2}+(1-{S}_{1}{)}^{2}=c$

${S}_{2}+(1-{S}_{1}{)}^{2}=c$

Now how to get the numerator integral, I'm not quite sure.

Also, is there any better, more elegant way of doing this problem? (I thought of using vectors, where the problem can be written as: $\overrightarrow{x}.\overrightarrow{1}=1$ then $P(||\overrightarrow{x}||<\sqrt{c})=?$. But then we would have to model the distribution of $||\overrightarrow{x}||$ which I have no idea about)

Or maybe a more direct approach? Like the ratio of areas (or volumes?).

asked 2022-08-16

Joint Probability of Independent Geometric Random Variables

Let $X,Y\sim G(p)$ be independent Geometric random variables ($p\in (0,1)$). Show that $P(X=Y)=p/(2-p)$.

Let $X,Y\sim G(p)$ be independent Geometric random variables ($p\in (0,1)$). Show that $P(X=Y)=p/(2-p)$.

asked 2022-08-18

Geometric probability - parallelograms

Inside a rhombus E with sides 10 unit and one interior angle less than 90 degree, there are 2 parallel (with E) parallelograms A and B, both can move freely and uniformly inside E but must keep parallel with E in moving. A is with base 8 unit and adjacent side 6 unit; while B is with base 5 unit and adjacent side 9 unit. If a point is chosen randomly in E, find the probability that the point lies inside A and B at the same time.

Inside a rhombus E with sides 10 unit and one interior angle less than 90 degree, there are 2 parallel (with E) parallelograms A and B, both can move freely and uniformly inside E but must keep parallel with E in moving. A is with base 8 unit and adjacent side 6 unit; while B is with base 5 unit and adjacent side 9 unit. If a point is chosen randomly in E, find the probability that the point lies inside A and B at the same time.

asked 2022-07-17

Mathematically prove that a Beta prior distribution is conjugate to a Geometric likelihood function

I have to prove with a simple example and a plot how prior beta distribution is conjugate to the geometric likelihood function. I know the basic definition as

'In Bayesian probability theory, a class of distribution of prior distribution θ is said to be the conjugate to a class of likelihood function $f(x|\theta )$ if the resulting posterior distribution is of the same class as of $f(\theta )$.'

But I don't know how to prove it mathematically.

I have to prove with a simple example and a plot how prior beta distribution is conjugate to the geometric likelihood function. I know the basic definition as

'In Bayesian probability theory, a class of distribution of prior distribution θ is said to be the conjugate to a class of likelihood function $f(x|\theta )$ if the resulting posterior distribution is of the same class as of $f(\theta )$.'

But I don't know how to prove it mathematically.

asked 2022-08-19

Branching Process - Extinction probability geometric

Consider a branching process with offspring distribution Geometric($\alpha $); that is, ${p}_{k}=\alpha (1-\alpha {)}^{k}$ for $k\ge 0$.

a) For what values of $\alpha \in (0,1)$ is the extinction probability $q=1$.

Let $\{{Z}_{n}{\}}_{n\ge 0}$ be a branching process with ${p}_{0}>0$. Let $\mu =\sum _{k\ge 0}k{p}_{k}$ be the mean of the offspring distribution and let $g(s)=\sum _{k\ge 0}{s}^{k}{p}_{k}$ be the probability generating function of the offspring distribution.

- If $\mu \le 1$, then the extinction probability $q=1$.

- If $\mu >1$, then the extinction probability q is the unique solution to the equation $s=g(s)$ with $s\in (0,1)$.

b) Use the following proposition to give a formula for the extinction probability of the branching process for any value of the parameter $\alpha \in (0,1)$.

Consider a branching process with offspring distribution Geometric($\alpha $); that is, ${p}_{k}=\alpha (1-\alpha {)}^{k}$ for $k\ge 0$.

a) For what values of $\alpha \in (0,1)$ is the extinction probability $q=1$.

Let $\{{Z}_{n}{\}}_{n\ge 0}$ be a branching process with ${p}_{0}>0$. Let $\mu =\sum _{k\ge 0}k{p}_{k}$ be the mean of the offspring distribution and let $g(s)=\sum _{k\ge 0}{s}^{k}{p}_{k}$ be the probability generating function of the offspring distribution.

- If $\mu \le 1$, then the extinction probability $q=1$.

- If $\mu >1$, then the extinction probability q is the unique solution to the equation $s=g(s)$ with $s\in (0,1)$.

b) Use the following proposition to give a formula for the extinction probability of the branching process for any value of the parameter $\alpha \in (0,1)$.