Further Stats Chapter 3: Geometric and Negative Binomial Distributions

10 flashcards · Mathematics · by TheoTyson

1. When is the geometric distribution used?

   To find the probability it takes a certain number of trials to get one success.

2. What are the criteria for the geometric distribution?

   The trials are successive. The trials are independent. The probability of success is the same for all trials.

3. What is the equation for the geometric distribution? X∼Geo(n,p)X \sim \text{Geo}(n, p)X∼Geo(n,p)

   P(X=x)=p(1−p)x−1P(X=x) = p(1-p)^{x-1}P(X=x)=p(1−p)x−1

4. What is the equation for the cumulative geometric distribution? X∼Geo(p)X \sim \text{Geo}(p)X∼Geo(p)

   P(X⩽x)=1−(1−p)xP(X \leqslant x) = 1 - (1 - p)^xP(X⩽x)=1−(1−p)x

5. What is the mean of X∼Geo(p)X \sim \text{Geo}(p)X∼Geo(p) ?

   E(X)=1pE(X) = \frac{1}{p}E(X)=p1​

6. What is the variance of X∼Geo(p)X \sim \text{Geo}(p)X∼Geo(p) ?

   E(X)=1−pp2E(X) = \frac{1-p}{p^2}E(X)=p21−p​

7. When is the negative binomial distribution used?

   To find the probability it takes a certain number of trials to get a certain number of successes.

8. What is the equation for the negative binomial distribution? X∼NB(r,p)X \sim \text{NB}(r, p)X∼NB(r,p)

   P(X=x)= x−1Cr−1 pr(1−p)x−rP(X = x) = \, ^{x-1}C_{r-1} \, p^r (1-p)^{x-r}P(X=x)=x−1Cr−1​pr(1−p)x−r r=r =r= number of successes x=x = x= number of trials

9. What is the mean of X∼NB(r,p)X \sim \text{NB}(r, p)X∼NB(r,p) ?

   E(X)=rpE(X) = \frac{r}{p}E(X)=pr​

10. What is the variance of X∼NB(r,p)X \sim \text{NB}(r, p)X∼NB(r,p) ?

    Var(X)=r(1−p)p2\text{Var}(X) = \frac{r(1-p)}{p^2}Var(X)=p2r(1−p)​