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Magazine Chapter 3: Problem 133
Show that the probability density function of a negative binomial random variable equals the probability density function of a geometric random variable when \(r=1\). Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for a geometric random variable when \(r=1\).
Short Answer
Expert verified
The PDFs and the mean and variance formulas are identical for both distributions when \( r=1 \).
Step by step solution
01
Identify the PDF of the Negative Binomial Distribution
The probability density function (PDF) of a negative binomial random variable is given by \( P(X=k) = \binom{k+r-1}{k}(1-p)^r p^k \), where \( r \) is the number of successes, \( p \) is the probability of success in a single trial, and \( k \) is the number of failures. We must show this is equal to the geometric distribution when \( r=1 \).
02
Substitute and Simplify for r=1
Substituting \( r=1 \) into the negative binomial PDF, we have: \( P(X=k) = \binom{k}{k}(1-p)p^k \). This simplifies to \( P(X=k) = (1-p)p^k \).
03
Identify the PDF of a Geometric Distribution
The probability density function for a geometric distribution is \( P(Y=k) = (1-p)p^k \). This distribution describes the number of Bernoulli trials needed to get one success with success probability \( p \).
04
Comparison of PDFs
By comparing \( P(X=k) = (1-p)p^k \) for the negative binomial with \( P(Y=k) = (1-p)p^k \) for the geometric distribution, we observe that they are identical when \( r=1 \).
05
Formulas for Mean and Variance of Negative Binomial
The mean of a negative binomial distribution is given by \( \frac{r(1-p)}{p} \) and the variance is \( \frac{r(1-p)}{p^2} \). Substituting \( r=1 \), the mean becomes \( \frac{1(1-p)}{p} = \frac{1-p}{p} \) and the variance becomes \( \frac{1(1-p)}{p^2} = \frac{1-p}{p^2} \).
06
Formulas for Mean and Variance of Geometric Distribution
For a geometric distribution, the mean is \( \frac{1-p}{p} \), and the variance is \( \frac{1-p}{p^2} \). These match exactly with the mean and variance derived for the negative binomial distribution when \( r=1 \).
07
Conclusion
When \( r=1 \), the probability density functions and the formulas for the mean and variance of negative binomial and geometric distributions are identical, confirming the required conditions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Density Function
The probability density function, or PDF, is a crucial component in understanding probability distributions. For the negative binomial distribution, the PDF describes the probability of achieving a certain number of failures before a specified number of successes occurs. In the formula \( P(X=k) = \binom{k+r-1}{k}(1-p)^r p^k \), each term has a specific role:
- \( k \) is the number of failures desired.
- \( r \) is the number of successes needed.
- \( p \) represents the probability of success on each trial.
Geometric Distribution
The geometric distribution is a specific case of the negative binomial distribution. It is employed for scenarios where one wants to count the number of Bernoulli trials needed to achieve the first success.
- The geometric PDF is expressed as \( P(Y=k) = (1-p)p^k \).
- It requires success on the first trial after \( k \) failures.
- The parameter \( p \) remains the probability of success per trial.
Mean and Variance
The concepts of mean and variance are pivotal when exploring probability distributions, like the negative binomial and geometric distributions. The mean indicates the expected number of trials, while the variance measures the spread or variability of the trials.
- For the negative binomial distribution, with mean \( \frac{r(1-p)}{p} \) and variance \( \frac{r(1-p)}{p^2} \), substituting \( r=1 \) simplifies these to geometric distribution鈥檚 mean \( \frac{1-p}{p} \) and variance \( \frac{1-p}{p^2} \).
- The calculations show that when \( r = 1 \), both distributions share the same statistical properties.
Negative Binomial Random Variable
A negative binomial random variable describes the number of trials required to achieve a specific number of successes. This variable becomes particularly interesting when \( r=1 \), effectively transforming it into a geometric random variable.
- The negative binomial distribution accommodates different numbers of successes, \( r \), offering flexibility.
- When \( r=1 \), the focus narrows to just achieving the first success (geometric distribution).
- This variable thus captures a comprehensive range of scenarios鈥攆rom just getting one success to a desired number of successes鈥攚ith the probability of each scenario neatly described through its distribution.
