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Questions and Answers
What is the sample space for the Negative Binomial Distribution?
What is the sample space for the Negative Binomial Distribution?
- S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n − 1; r of w1 , w2 ,. , wn−1 are s and remaining n − r of w1 , w2 ,. , wn−1 are f }
- S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n; r − 1 of w1 , w2 ,. , wn−1 are s and remaining n − r of w1 , w2 ,. , wn−1 are f }
- S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n; r of w1 , w2 ,. , wn are s and remaining n − r of w1 , w2 ,. , wn are f }
- S = {(w1 , w2 ,. , wn ) : n ∈ {r, r + 1,.}, wn = s, wi ∈ {s, f }, i = 1, 2,. , n − 1; r − 1 of w1 , w2 ,. , wn−1 are s and remaining n − r of w1 , w2 ,. , wn−1 are f } (correct)
Which of the following best describes the Negative Binomial Distribution?
Which of the following best describes the Negative Binomial Distribution?
- A distribution that models the number of failures before the r-th success in a sequence of independent Bernoulli trials (correct)
- A distribution that models the number of successes in a fixed number of independent Bernoulli trials
- A distribution that models the number of successes before the r-th failure in a sequence of independent Bernoulli trials
- A distribution that models the number of failures in a fixed number of independent Bernoulli trials
What does the random variable X represent in the Negative Binomial Distribution?
What does the random variable X represent in the Negative Binomial Distribution?
- The number of failures in a fixed number of independent Bernoulli trials
- The number of failures before the r-th success (correct)
- The number of successes before the r-th failure
- The number of successes in a fixed number of independent Bernoulli trials
What is the probability of the event {X = x} for x ∈ {0, 1, 2, · · · } in the Negative Binomial Distribution?
What is the probability of the event {X = x} for x ∈ {0, 1, 2, · · · } in the Negative Binomial Distribution?
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What does the probability p represent in the Negative Binomial Distribution?
What does the probability p represent in the Negative Binomial Distribution?
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Study Notes
Sample Space of Negative Binomial Distribution
- The sample space consists of non-negative integers: {0, 1, 2, ...} representing the number of failures before achieving a specified number of successes.
Description of Negative Binomial Distribution
- The Negative Binomial Distribution models the number of failures in a sequence of Bernoulli trials before a predetermined number of successes occurs.
- It is a generalization of the geometric distribution, which focuses on the number of trials until the first success.
Random Variable X in Negative Binomial Distribution
- The random variable X represents the total number of failures encountered before achieving a specified number of successful trials (r).
Probability of Event {X = x}
- The probability mass function for the Negative Binomial Distribution is given by:
- P(X = x) = C(x + r - 1, r - 1) * p^r * (1 - p)^x
- Where C indicates the binomial coefficient, p is the probability of success on each trial, and x is the number of failures.
Probability p in Negative Binomial Distribution
- The probability p denotes the likelihood of success in each individual Bernoulli trial.
- It is a key parameter and influences the shape of the distribution.
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Description
Test your understanding of the Negative Binomial and Geometric distributions with this quiz. Explore the concepts of success probability and the number of trials needed for a certain number of successes.
