Discrete random variables：練習題

Given that \(X \sim B(50, 0.2)\), find the variance of \(X\).

The random variable \(X\) represents the number of heads obtained when three biased coins are tossed. For each coin, the probability of obtaining a head is \(\frac{2}{3}\). Find \(P(X \ge 2)\).

If the discrete random variable \(X\) follows a binomial distribution \(B(10, 0.3)\), what is the mean of \(X\)?

A binomial random variable \(X \sim B(n, p)\) has a mean of \(4\) and a variance of \(2.4\). Determine the value of the parameter \(n\).

Which of the following functions could represent a valid probability distribution for a discrete random variable \(X\) taking values in the set \(\{1, 2, 3\}\)?

State two necessary conditions for a random variable to be modeled accurately by a binomial distribution.

The probability distribution of \(X\) is given by \(P(X=1) = a\), \(P(X=2) = 0.4\), and \(P(X=3) = 0.6 - a\). Find the mean of \(X\) in terms of \(a\).

The probability that a certain type of seed germinates is \( 0.8 \). If \( 10 \) such seeds are planted independently, calculate the probability that exactly \( 8 \) seeds germinate, leaving your answer in terms of powers.

The discrete random variable \(X\) has mean \(E(X) = 5\) and variance \(Var(X) = 3\). Calculate the following:
(a) \(E(4X - 2)\)
(b) \(Var(4X - 2)\)

A fair six-sided die is rolled 8 times. A player wins \$10 if a '6' is rolled and loses \$2 for any other outcome. Let \(X\) be the number of '6's obtained in the 8 rolls.
(a) State the distribution of \(X\), including its parameters.
(b) Express the total profit \(W\) in terms of \(X\).
(c) Calculate the expected total profit for the player.