Bernoulli and binomial distributions：練習問題

A Bernoulli random variable \( X \) has a success probability \( p = 0.4 \). Find the variance of \( X \).

A binomial random variable \(X\) follows the distribution \(B(n, p)\). Given that the mean of \(X\) is 4 and the variance is 2.4, determine the number of trials \(n\).

A machine produces components with a success probability of \(p=0.2\). Find the minimum number of independent trials \(n\) required such that the probability of at least one success is greater than 0.9.

Let \(X\) be a discrete random variable such that \(X \sim B(4, p)\). If the probability of zero successes is 0.0625, find the expected value \(E(X)\).

For a discrete random variable \(X \sim B(n, p)\), which of the following expressions correctly represents the ratio of consecutive probabilities \(\frac{P(X = k)}{P(X = k-1)}\) for \(1 \le k \le n\)?

Define the construction of a binomial random variable \(X\) in terms of Bernoulli trials and state the specific conditions required for the trials.

Using the mean \(E(X) = np\) and variance \(Var(X) = np(1 - p)\) for \(X \sim B(n, p)\), derive an expression for \(E(X^2)\) in terms of \(n\) and \(p\).

Explain the relationship between the variance of a Bernoulli trial and its probability of success \(p\), and determine the value of \(p\) that maximizes this variance.

The random variable \(X\) follows a binomial distribution such that \(X \sim B(8, 0.25)\).

(a) Find the probability that \(X = 2\).
(b) Find the probability that \(2 \le X < 4\).
(c) State the mean and variance of \(X\).

A binomial random variable \(X\) is such that \(E(X) = 15\) and \(Var(X) = 3.75\).

(a) Find the values of the parameters \(n\) and \(p\).
(b) Calculate the probability \(P(X = n)\), giving your answer in scientific notation to 3 significant figures.
(c) Explain why a Bernoulli distribution is a special case of this distribution.