For all questions that involve the geometric distribution in the context of probability, that is, the law of a random variable whose outcome is the number of attempts we need before a first success in repeated Bernoulli experiments.
Questions tagged [geometric-distribution]
73 questions
16
votes
2 answers
Characteristic function of exponential and geometric distributions
I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you please guide me on getting them?
Here is my solution so far:
Characteristic function of exponential distribution:
$\phi(t) =…
user48405
5
votes
2 answers
If population keep trying till they have girl child, what will be the probability of population having more girls than boys and vice versa?
I was solving this problem:
In a world where everyone wants a girl child, each family continues having babies till they have a girl. What do you think will the boy to girl ratio be eventually? (Assuming probability of having a boy or a girl is the…
Mahesha999
- 2,081
4
votes
3 answers
How to dither binned data (following a geometric distribution) to recover the exponential distribution?
Consider a random variable that follows an exponential distribution. After binning (floor), becomes discrete and follows a geometric distribution. My question is: how can we recover the original exponential distribution by adding random noise to…
Roland
- 143
- 6
4
votes
2 answers
Prove $\sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} {n \choose k} p^k (1-p)^{n-k}$
Prove that the following two summations are equal for any positive integers $i\leq n$, and any real number $p$ between $0$ and $1$:
$$
\sum_{k=i}^{n} {k-1 \choose i-1} p^i (1-p)^{k-i} = \sum_{k=i}^{n} {n \choose k} p^k (1-p)^{n-k}
$$
I know the…
Yuzhen Feng
- 95
4
votes
2 answers
Find all $f$ such that $X\sim\mathcal{G}(\lambda) \;\Rightarrow\; f(X)\sim \mathcal{G}(\mu)$
I found a nice problem recently, but could not come up with a solution:
Find all functions $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ such that for all $0< \lambda < 1$, if $X\sim G(\lambda)$, then there exists $0<\mu < 1$ with $f(X) \sim…
Noomkwah
- 618
4
votes
2 answers
Expectation and variance of the geometric distribution
How can one use memorylessness and the law of total expectation and the law of total variance to find the expectation and variance of the geometric distribution?
I will post my own answer, but as always, that shouldn't stop anyone else from posting…
4
votes
1 answer
Questions about geometric distribution
I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd Johnson, Adrienne W. Kemp, Samuel Kotz.
The…
Tim
- 49,162
3
votes
0 answers
Are $X - Y$ and $\min(X, Y)$ independent for independent geometric random variables $X$ and $Y$?
Question:
If $X$ and $Y$ are two independent geometric random variables with parameters $a$ and $b$, where
$$\forall k \geq 1, P(X = k) = a(1-a)^{k-1} $$
and
$$\forall k \geq 1, P(Y = k) = b(1-b)^{k-1}$$
Then are $X - Y$ and $\min(X, Y)$…
3
votes
3 answers
Find Variance of Geometric Random Variable Using Law of Total Expectation
I'm trying to compute variance of geometric RV $X$ with parameter p. I would like to use the Law of Total Expectation. RV $Y$ represents the first trial, which is either success with probability $p$ or fail with probability $(1-p)$.
$
\text{Var}(X)…
Filip
- 59
3
votes
0 answers
Geometric Distribution in Simple Random Walk
My question is related to this question here.
This says that if $S_{n}$ is a simple random walk (with steps $+1$ or $-1$ with probability $p$ and $q$ respectively) started at $S_{0}=1$, and if $T=\min\{k\geq 0:S_{k}=0\}$ then $$Z_k = \sum…
Blitzkrieg
- 243
3
votes
1 answer
Geometric Random Variable Expectation/Variance
Looking to find the expected value/variance of the following geometric variable defined as having success rate p, where trials are taken until we have a success (which happens with probability p) or we have taken n trials.
Successfully calculated…
John Li
- 157
3
votes
0 answers
Memoryless property of the discrete geometric distribution
I came across this proof of the memoryless property of the discrete geometric distribution which defines the criterion of memoryless as:
\begin{equation}\label{memoryless-def}
\boldsymbol{\operatorname{P}}(X \ge s+t | X \ge t) =…
3
votes
1 answer
On the distribution of the minimum of geometrically distributed random variables
I am struggling with the following homework assignment. I have been on this for an hour and have nothing:
Let $X_1, \dots X_n$ be independent random variables with geometric distribution, $X_i \sim \operatorname{Geom}(p_i)$. Let $Y := \min \{X_1 ,…
oopsi
- 165
2
votes
1 answer
Asymptotic distribution of MLE of geometric distribution
I need to find the asymptotic distribution of the MLE of a geometric distribution.
I know $\overline X$ goes as $N(1/p, (1-p)/(n p^2))$. Using the delta method MLE=$1/\overline X$ goes as $N(p, (1-p)/(np^6))$.
However if I use the asymptotic theory…
2
votes
1 answer
CDF of a distribution defined for all integers
Some time back, I asked a question regarding a mass function for a dice roll with the following rules:
We roll $n$ fair, $d$-sided dice (with standard number $1$ to $d$), choosing target numbers $x$ and $y$, such that $d \geq x > y \geq 1$, and…
