Geometric and Binomial Distributions

Front

What are the assumptions of the Binomial distribution?

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1. All trials must be independent. 2. Each trial must be classified as a success or a failure. 3. The number of trials is fixed. 4. The probability of success is constant across trials.

Front

What is a continuous random variable?

Back

A continuous random variable can take any value within a given range. For example, the time it takes a student to complete an exam is a continuous random variable.

Front

What is the probability of exactly k successes in n trials in a Binomial distribution?

Back

The probability is given by the formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success.

Front

What is the expected value of a Binomial distribution?

Back

The expected value (mean) is calculated as E(X) = n * p, where n is the number of trials and p is the probability of success.

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What is the difference between Binomial and Geometric distributions?

Back

Binomial distribution counts the number of successes in a fixed number of trials, while Geometric distribution counts the number of trials until the first success.

Front

What is the formula for the probability of rolling a 6 for the first time on the k-th roll of a die?

Back

P(X = k) = (1/6) * (5/6)^(k-1), where k is the number of rolls.

Front

How do you calculate the expected number of trials until the first success in a Geometric distribution?

Back

The expected number of trials is given by E(X) = 1/p, where p is the probability of success.

Front

What is the variance of a Binomial distribution?

Back

The variance is calculated as Var(X) = n * p * (1-p), where n is the number of trials and p is the probability of success.

Front

What is the probability of getting at least one success in n trials?

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P(at least one success) = 1 - P(no successes) = 1 - (1-p)^n.

Front

What is the cumulative distribution function (CDF) for a Binomial distribution?

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The CDF gives the probability of obtaining k or fewer successes in n trials: P(X ≤ k) = Σ (n choose i) * p^i * (1-p)^(n-i) for i = 0 to k.