概率與統計中的數學模型：練習題

A quality control manager inspects batches of 20 electronic components. It is known that 3% of all components produced are defective. Assuming the components are selected randomly and independently, which probability distribution is the most appropriate model for the number of defective components, \(X\), found in a batch of 20?

The heights of adult men in a certain population are modelled by a Normal distribution with a mean of \(178 \text{ cm}\) and a standard deviation of \(6 \text{ cm}\). If a man is chosen at random, find the probability that his height is less than \(184 \text{ cm}\).

In a large batch of 500 items, the probability that any single item is defective is \(0.006\). Let \(Y\) be the number of defective items in the batch. Using a suitable approximation, estimate the probability that there are exactly 3 defective items.

A call centre receives telephone calls at a mean rate of 4.5 calls per 10-minute period. It is assumed that the calls occur randomly and independently. Let \(X\) be the number of calls received in a randomly selected 20-minute period. Calculate \(P(X = 7)\).

The time, \(T\) minutes, a customer waits for a service is modelled by a Continuous Uniform distribution over the interval \([0, 8]\). A customer is chosen at random. Given that the customer waited for at least 3 minutes, find the probability that the customer waited for less than 6 minutes. State a necessary assumption required for this continuous uniform model to be appropriate.

A discrete random variable \(Y\) follows a discrete uniform distribution over the set of integers \(\{1, 2, 3, ..., n\}\).
(a) Given that \(E(Y) = 8.5\), find the value of \(n\).
(b) For this value of \(n\), calculate \(Var(Y)\).
(c) A second random variable \(W\) is defined such that \(W = 10 - 2Y\). Find \(P(W < 0)\).