Which of the following mentioned standard probability density functions is applicable to discrete random variables?

General
1

which of the following mentioned standard probability density functions is applicable to discrete random variables?

2

Which of the following mentioned standard probability density functions is applicable to discrete random variables?

In probability theory and statistics, probability distributions for discrete random variables are often described using probability mass functions (PMFs) rather than probability density functions (PDFs), which are more applicable to continuous random variables. Discrete random variables take on distinct, separate values, often integers, and each outcome has a specific probability associated to it.

Here are some common discrete probability distributions:

1. Binomial Distribution

  • Definition: The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials, with a constant probability of success in each trial.
  • Parameters: Number of trials n and probability of success p per trial.
  • Probability Mass Function (PMF):
    P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
    where k is the number of successes.

2. Poisson Distribution

  • Definition: The Poisson distribution models the number of times an event occurs in a fixed interval of time or space, given the events occur with a known constant mean rate and independently of the time since the last event.
  • Parameter: Average rate \lambda (lambda) of occurrence.
  • Probability Mass Function (PMF):
    P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
    where k is a non-negative integer representing the number of occurrences.

3. Geometric Distribution

  • Definition: The geometric distribution models the number of trials needed for the first success in a series of independent and identical Bernoulli trials.
  • Parameter: Probability of success p on each trial.
  • Probability Mass Function (PMF):
    P(X = k) = (1-p)^{k-1}p
    where k is the trial on which the first success occurs.

4. Negative Binomial Distribution

  • Definition: The negative binomial distribution extends the geometric distribution to model the number of trials required to achieve a specified number of successes.
  • Parameters: Number of successes r and probability of success p in each trial.
  • Probability Mass Function (PMF):
    P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}
    where k is the total number of trials.

5. Hypergeometric Distribution

  • Definition: The hypergeometric distribution gives the probability of k successes (draws of the first type) in n draws, without replacement, from a finite population of size N, that contains exactly K successes.
  • Parameters: Population size N, number of successes in the population K, number of draws n.
  • Probability Mass Function (PMF):
    P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}

Conclusion

From this evaluation, we confirm that while continuous random variables are associated with probability density functions (PDFs), discrete random variables utilize probability mass functions (PMFs). Some popular distributions applicable to discrete random variables include the Binomial, Poisson, Geometric, Negative Binomial, and Hypergeometric distributions. Each of these distributions describes differing scenarios and data properties relating to distinct random variables.

If you want to dive deeper into any of these distributions or need examples demonstrating their application, feel free to let me know! @LectureNotes