Suppose X is a binomial random variable. To approximate P(X < 5), compute ________.

Verified step by step guidance

1. Identify the parameters of the binomial random variable \(X\), namely the number of trials \(n\) and the probability of success \(p\) for each trial.
2. Since \(X\) is binomial, recall that \(X \sim \text{Binomial}(n, p)\), and the exact probability \(P(X < 5)\) means \(P(X \leq 4)\), which is the sum of probabilities from \(X=0\) to \(X=4\).
3. To approximate \(P(X < 5)\), consider using a normal approximation to the binomial distribution if \(n\) is large and \(p\) is not too close to 0 or 1. The normal approximation uses a normal distribution with mean \(\mu = np\) and variance \(\sigma^2 = np(1-p)\).
4. Apply the continuity correction by approximating \(P(X < 5)\) with \(P(Y < 4.5)\) where \(Y\) is the normal random variable \(Y \sim N(\mu, \sigma^2)\).
5. Standardize the value 4.5 to find the corresponding \(z\)-score using the formula \(z = \frac{4.5 - \mu}{\sigma}\), then use the standard normal distribution table or a calculator to find \(P(Z < z)\), which approximates \(P(X < 5)\).

Key Concepts

Binomial Random Variable

A binomial random variable represents the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by parameters n (number of trials) and p (probability of success). Understanding its distribution is essential for calculating probabilities like P(X < 5).

Normal Approximation to the Binomial

When the number of trials n is large, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This simplifies probability calculations, especially for cumulative probabilities like P(X < 5), by using the normal distribution instead of binomial formulas.

Continuity Correction

Since the binomial distribution is discrete and the normal distribution is continuous, a continuity correction adjusts for this difference by adding or subtracting 0.5 to the discrete x-value when approximating probabilities. For P(X < 5), we use P(X < 4.5) in the normal approximation to improve accuracy.