A) $P\{X+Y=n\}(n=1,2,...)$?

Jaxon Hamilton
2022-07-18
Answered

X and Y are geometric RV's with parameter p.

A) $P\{X+Y=n\}(n=1,2,...)$?

A) $P\{X+Y=n\}(n=1,2,...)$?

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asked 2022-08-09

Probability of Binomial twice of Geometric

I've come up with an interesting result:

Let X be the amount of failures of Bernoulli(p) until we get (p).

$X=Geo(p)$

$B=Bin(2X,\frac{1}{2})$

$P(B=X)=\text{}?$

Turns out: $P(B=X)=\sqrt{p}$

I found it using the Taylor expansion of $\frac{1}{\sqrt{1-p}}$, where the coefficient of ${p}^{i}$ turns out to be $P(Bin(2i,\frac{1}{2})=i)$.

I would like to see a probabilistic proof of this result.

Explanation of the process in words:

Roll a die with probability p of getting "X". Each time that we don't get "X", toss 2 balanced coins and accumulate the number of heads and tails. When you get "X", check if you got the same amount of heads and tails.

Programmatic explanation:

$i=0$

$while(!bernuli(p))\text{}\text{i++;}$

$Bin(2i,\frac{1}{2})\stackrel{?}{=}i$

Example:

If we succeed immediately (with probability p), $X=0$, and $P(Bin(2\cdot 0,\frac{1}{2})=0)=1$, thus $P(B=0)=1$, thus contributing p to the conditional sum, $p<\sqrt{p}$, and everything is alright.

I've come up with an interesting result:

Let X be the amount of failures of Bernoulli(p) until we get (p).

$X=Geo(p)$

$B=Bin(2X,\frac{1}{2})$

$P(B=X)=\text{}?$

Turns out: $P(B=X)=\sqrt{p}$

I found it using the Taylor expansion of $\frac{1}{\sqrt{1-p}}$, where the coefficient of ${p}^{i}$ turns out to be $P(Bin(2i,\frac{1}{2})=i)$.

I would like to see a probabilistic proof of this result.

Explanation of the process in words:

Roll a die with probability p of getting "X". Each time that we don't get "X", toss 2 balanced coins and accumulate the number of heads and tails. When you get "X", check if you got the same amount of heads and tails.

Programmatic explanation:

$i=0$

$while(!bernuli(p))\text{}\text{i++;}$

$Bin(2i,\frac{1}{2})\stackrel{?}{=}i$

Example:

If we succeed immediately (with probability p), $X=0$, and $P(Bin(2\cdot 0,\frac{1}{2})=0)=1$, thus $P(B=0)=1$, thus contributing p to the conditional sum, $p<\sqrt{p}$, and everything is alright.

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 2022-07-21

A geometrically distributed random variable where the first success probability is different

Your first success probability ${p}_{a}$ is higher on the first trial, than on the remaining trials (where it is ${p}_{b}$, constantly). How does this impact the mean of the general experiment (1/p in the general, fixed geometric case) and the standard deviation $\sqrt{(1-p)/{p}^{2}}$ in the general, fixed geometric case)?

I think I got the mean:

${p}_{a}+(1-{p}_{a})(1/{p}_{b}+1)$

Your first success probability ${p}_{a}$ is higher on the first trial, than on the remaining trials (where it is ${p}_{b}$, constantly). How does this impact the mean of the general experiment (1/p in the general, fixed geometric case) and the standard deviation $\sqrt{(1-p)/{p}^{2}}$ in the general, fixed geometric case)?

I think I got the mean:

${p}_{a}+(1-{p}_{a})(1/{p}_{b}+1)$

asked 2022-08-03

Let $\lambda$ be a positive real number. Consider a semicircle of center O and diameter AB. Choose points C and D (C is between D and B) on the semicircle and let \angle AOD=2\alpha and \angle BOC=2\beta.

The point X on the line CD is such that XD/XC=\lambda. Prove that when \alpha and \beta satisfy \tan\alpha=\tan\beta+\sqrt{3}/2, all lines through X perpendicular to CD pass through a fixed point.

asked 2022-07-17

A, B, and C Roll Dice

Three players A, B, and C take turns rolling a pair of dice. The winner is the first player who obtains the sum of 7 $(P[7]=1/6)$ on a given roll of the dice. If A rolls 1st, then B, then C, then back to A (etc. etc. etc.). What is the P[A wins], P[B wins], P[C wins]?

Three players A, B, and C take turns rolling a pair of dice. The winner is the first player who obtains the sum of 7 $(P[7]=1/6)$ on a given roll of the dice. If A rolls 1st, then B, then C, then back to A (etc. etc. etc.). What is the P[A wins], P[B wins], P[C wins]?

asked 2022-08-25

n points are chosen randomly from a circumference of a circle. Find the probability that all points are on the same half of the circle. My intuition was that the probability is $1/{2}^{n-1}$ since if the first point is placed somewhere on the circumference, for each point there is probability of 1/2 to be placed on the half to the right of it and 1/2 to the left..but i feel i might be counting some probabilitys twice.

asked 2022-09-26

Finding probability distribution of quantity depending on other distributions

I have a vector that depends on the coordinates of randomly drawn unit vectors in ${\mathbb{R}}^{2}$:

$\sigma =\left(\begin{array}{c}\frac{\mathrm{cos}({\theta}_{a})+\mathrm{cos}({\theta}_{b})}{1+\mathrm{cos}({\theta}_{a})\mathrm{cos}({\theta}_{b})}\\ \frac{\mathrm{sin}({\theta}_{a})+\mathrm{sin}({\theta}_{b})}{1+\mathrm{sin}({\theta}_{a})\mathrm{sin}({\theta}_{b})}\end{array}\right)$

Here, ${\theta}_{a},{\theta}_{b}\in [0,2\pi )$, are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in ${\mathbb{R}}^{2}$. I am interested in figuring out a probability distribution for the coordinates of the $\sigma $-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.

My questions are thus:

Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?

If the method exists, how would I use it on my concrete example?

I have a vector that depends on the coordinates of randomly drawn unit vectors in ${\mathbb{R}}^{2}$:

$\sigma =\left(\begin{array}{c}\frac{\mathrm{cos}({\theta}_{a})+\mathrm{cos}({\theta}_{b})}{1+\mathrm{cos}({\theta}_{a})\mathrm{cos}({\theta}_{b})}\\ \frac{\mathrm{sin}({\theta}_{a})+\mathrm{sin}({\theta}_{b})}{1+\mathrm{sin}({\theta}_{a})\mathrm{sin}({\theta}_{b})}\end{array}\right)$

Here, ${\theta}_{a},{\theta}_{b}\in [0,2\pi )$, are drawn uniformly random from the unit circle, and are thus angles that parametrize unit vectors in ${\mathbb{R}}^{2}$. I am interested in figuring out a probability distribution for the coordinates of the $\sigma $-vector, but I am not well-versed in probability theory. I have been told that it is possible to somehow find a probability distribution for a quantity that depends on other distributions by somehow using the Jacobian matrix, but I have been unable to figure out how on my own.

My questions are thus:

Is there a method for finding a probability distribution for a quantity that depends on other distributions? If so, how? Does it have a name so I can learn about form other sources?

If the method exists, how would I use it on my concrete example?