Bernoulli Distribution Quiz

Study Notes

Distributions in a Nutshell

Bernoulli Distribution: A discrete probability distribution describing a Bernoulli trial (an experiment with only two outcomes: success or failure).
Parameters: p (probability of success)
Range: x = 0, 1
Mean: p
Variance: pq
Binomial Distribution: Describes the number of successes in a fixed number of independent Bernoulli trials.
Parameters: n (number of trials), p (probability of success in a single trial)
Range: x = 0, 1, 2, ..., n
Mean: np
Variance: npq
Poisson Distribution: Describes the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.
Parameter: λ (average rate of events)
Range: x = 0, 1, 2, ... , ∞
Mean: λ
Variance: λ
Hypergeometric Distribution: Describes the probability of getting exactly k successes in n draws, without replacement, from a finite population of size N containing exactly K successes.
Parameters: N (population size), K (number of successes in the population), n (number of draws)
Range: x = min(a, n),..., max(a, n)
Mean: na/(a+b)
Variance: nab(a + b- n)/(a+b)²(a + b -1)
Normal Distribution: A continuous probability distribution with a symmetrical bell-shaped curve.
Parameters: μ (mean), σ (standard deviation)
Range: -∞ < x < ∞
Mean: μ
Variance: σ²
Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.
Parameters: 0 & 1
Range: -∞ < z < ∞
Mean: 0
Variance: 1
Chi-Square Distribution: A continuous probability distribution that arises frequently in statistical tests.
Parameter: n (degrees of freedom)
Range: 0 ≤ x² < ∞
Mean: n
Variance: 2n
Student's t-Distribution: A continuous probability distribution that is similar to the standard normal distribution but has heavier tails.
Parameter: n (degrees of freedom)
Range: -∞ < t < ∞
Mean: 0
Variance: n/(n - 2) for n > 2

Bernoulli Distribution Exercise

1. Bernoulli Variate: A random variable that takes on the value 1 (for success) or 0 (for failure) with probability p and 1-p, respectively, in a single Bernoulli trial.
2. Bernoulli Distribution: A probability distribution describing a Bernoulli trial.
3. Probability Mass Function: P(X=x) = p^x (1-p)^1-x, where x = 0 or 1
4. Range and Parameter: Range: 0, 1 Parameter: p (probability of success)
5. Example for Bernoulli Variate: Flipping a coin, where 1 represents heads and 0 represents tails.
6. Mean-Variance Relationship: Mean = p Variance = pq
7. Mean and Variance of Bernoulli Distribution: Mean = p, Variance = pq
8. Bernoulli Trial: A single trial that has only two possible outcomes: success or failure. Example: Flipping a coin; observing if a machine produces a defective part.
9. Distribution of Sum of Independent Bernoulli Variables: A binomial distribution
10. Bernoulli Distribution with p=0.23: A distribution where X = 0 with a probability of 0.77 and X = 1 with a probability of 0.23.

Description

Test your knowledge on the Bernoulli distribution with this quiz! Answer questions about parameters, mean, variance, and examples of Bernoulli trials. Challenge yourself and see how well you understand this fundamental concept in probability.