Bernoulli Distribution Quiz

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.