Common probability distributions including binomial, normal, and Poisson, with their defining properties and typical uses.
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- What is the binomial distribution used to model?
- The number of successes in a fixed number of independent trials, each with the same probability of success.
- What are the two parameters of a binomial distribution?
- n (number of trials) and p (probability of success on each trial).
- What is the probability mass function of a binomial distribution?
- P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) is the binomial coefficient.
- What is the mean of a binomial distribution with parameters n and p?
- np
- What is the variance of a binomial distribution with parameters n and p?
- np(1-p)
- Give a real-world example of a binomial distribution.
- The number of heads in 20 coin flips, or the number of defective items in a batch of 100 products.
- When n is large and p is small, the binomial distribution can be approximated by which distribution?
- The Poisson distribution.
- What is the range of possible values for X in a binomial distribution?
- X can be 0, 1, 2, ..., n (integer values from 0 to n).
- What is the normal distribution also called?
- The Gaussian distribution.
- What are the two parameters of a normal distribution?
- mu (mean) and sigma (standard deviation).
- What is the mean of the standard normal distribution?
- 0
- What is the standard deviation of the standard normal distribution?
- 1
- What is the probability density function for a normal distribution?
- f(x) = (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2/(2*sigma^2))
- What percentage of data falls within one standard deviation of the mean in a normal distribution?
- Approximately 68%.
- What percentage of data falls within two standard deviations of the mean in a normal distribution?
- Approximately 95%.