Final answer:
The cumulative distribution function of a binomial random variable with n = 3 and p = 1/4 is calculated by summing up the probabilities for each number of successes up to the specified point, using binomial probabilities and giving the answers as simplified fractions.
Step-by-step explanation:
To evaluate the cumulative distribution function (CDF) of a binomial random variable with n = 3 and p = 1/4 at specified points, we first need to understand the binomial distribution properties. For a given binomial random variable X, with the number of trials n and the probability of success p, the CDF is calculated by summing up the probabilities P(X = 0), P(X = 1), ..., up to P(X = x), where x is the specified point.
Here is a step-by-step calculation for each specified point (x):
- For x = 0 (no successes):
P(X = 0) = (3 choose 0) × (1/4)^0 × (3/4)^3 = 27/64 - For x = 1 (one success):
P(X = 0) + P(X = 1) = 27/64 + (3 choose 1) × (1/4)^1 × (3/4)^2 = 27/64 + 27/64 = 27/32 - For x = 2 (two successes):
P(X ≤ 1) + P(X = 2) = 27/32 + (3 choose 2) × (1/4)^2 × (3/4)^1 = 27/32 + 9/64 = 81/64 - For x = 3 (three successes):
P(X ≤ 2) + P(X = 3) = 81/64 + (3 choose 3) × (1/4)^3 × (3/4)^0 = 81/64 + 1/64 = 82/64 = 41/32
The calculated cumulative probabilities for each specified point give us the exact fractions that represent the CDF at those points. The probabilities for each X value can be calculated using the formula for the binomial probability:
P(X = x) = (n choose x) × p^x × (1-p)^(n-x). Remember to simplify your fractions to get the exact answers.