Question

Let the independent random variables \(X_{1}, X_{2}, \ldots, X_{40}\) be iid with the common pdf \(f(x)=3 x^{2}, 0

Short answer

The probability can be obtained via the formula for the binomial distribution and calculations of the series for k from 35 to 40. Final value requires computation based on the provided steps.

Step-by-step solution

Determine the single case probability

Firstly, we need to compute the individual probability of one random variable \(X_{i}\) exceeding 1/2. That means integrating the given pdf from 1/2 to 1. Thus we get:\[P1 = \int_{1/2}^{1} f(x) dx = \int_{1/2}^{1} 3x^{2} dx\]

Calculate the integral

Calculate the value of the integral to get the single-event probability. Substituting \(x^{2}\) with \(u\), we will have:\[P1 = [x^{3}]_{1/2}^{1} = 1^{3} - (\frac{1}{2})^{3} = 1 - \frac{1}{8} = \frac{7}{8}\]

Use the binomial distribution formula

Now we need to find the probability that at least 35 (from 35 to 40) of the variables exceed 1/2, which is a binomial distribution problem. The formula for binomial distribution is:\[C(n, k) * p^{k} * (1 - p)^{n-k}\]where n is the total number of trials, k is the number of successful trials, p is the probability of success and \(C(n, k)\) is the binomial coefficient standing for 'n choose k'. Here, n equals 40, p equals \(P1 = \frac{7}{8}\), and k ranges from 35 to 40. Hence, the required probability is the sum of the probabilities for k equals 35 to 40.

Calculate the total probability

Using the binomial distribution formula, calculate the sum of probabilities for k from 35 to 40:\[P = \sum_{k=35}^{40} C(40, k) * (\frac{7}{8})^{k} * (1 - \frac{7}{8})^{40-k}\]Compute this expression to get the final answer.