In a binomial experiment with n trials and probability of success p, if __ ________, the binomial random variable X is approximately normal with μX = ________ and σX = ________.

Verified step by step guidance

1. Identify the conditions under which a binomial distribution can be approximated by a normal distribution. Typically, this approximation is valid when both the expected number of successes and failures are sufficiently large, often stated as \(n p \geq 10\) and \(n (1 - p) \geq 10\).
2. Recall the mean (expected value) of a binomial random variable \(X\) with parameters \(n\) and \(p\) is given by the formula: \(\mu_X = n p\).
3. Recall the standard deviation of the binomial random variable \(X\) is given by the formula: \(\sigma_X = \sqrt{n p (1 - p)}\).
4. Understand that when the conditions for normal approximation are met, the binomial distribution \(X\) can be approximated by a normal distribution with mean \(\mu_X\) and standard deviation \(\sigma_X\) as calculated above.
5. Summarize the result: If \(n p \geq 10\) and \(n (1 - p) \geq 10\), then \(X \sim \text{Binomial}(n, p)\) is approximately \(N(\mu_X, \sigma_X^2)\) where \(\mu_X = n p\) and \(\sigma_X = \sqrt{n p (1 - p)}\).

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. It is discrete and defined by parameters n (number of trials) and p (probability of success).

Normal Approximation to the Binomial

When the number of trials n is large and both np and n(1-p) are greater than or equal to 10, the binomial distribution can be approximated by a normal distribution. This simplifies calculations by using continuous probability methods.

Mean and Standard Deviation of Binomial Distribution

The mean (μX) of a binomial random variable X is given by np, representing the expected number of successes. The standard deviation (σX) is √(np(1-p)), measuring the variability around the mean.