Geometric Distributions: Counting Conventions and Waiting-Time Formulas

All flashcards

What is the probability that the first success occurs on trial $x$ when $X$ counts trials through success?

$P(X=x)=(1-p)^{x-1}p$ for $x=1,2,3,\ldots$.

How are the trial-count variable $X$ and failure-count variable $Y$ related?

$X=Y+1$, because $X$ includes the successful trial and $Y$ counts only earlier failures.

What are the mean and standard deviation of the number of trials through the first success?

$\mu_X=1/p$ and $\sigma_X=\sqrt{1-p}/p$.

What is the common ratio of successive point probabilities in a geometric trial-count distribution?

$1-p$; each next point probability is the previous one multiplied by $1-p$.

What does the memoryless property say about a geometric waiting time?

Earlier failures do not change the distribution of the remaining wait for success.

Quiz questions

1. If $X$ counts trials through the first success, what is $P(X=x)$ for $x\geq 1$?Answer: $(1-p)^{x-1}p$
2. For a geometric variable $X$ counting trials through the first success, what is its expected value?Answer: $1/p$
3. Which equation expresses memorylessness for a trial-count geometric variable, for integers $n\geq1$ and $k\geq0$?Answer: $P(X>k+n\mid X>k)=P(X>n)$
4. For $X$ counting trials through the first success, what is the ratio $P(X=x+1)/P(X=x)$?Answer: $1-p$
5. If $Y$ counts failures before the first success, which probability formula applies for $y\geq0$?Answer: $P(Y=y)=(1-p)^y p$