Discrete random variables：練習問題

A discrete random variable \(X\) has the probability distribution given by \(P(X = x) = \frac{x}{k}\) for \(x = 1, 2, 3, 4\).
Find the value of the constant \(k\).

A discrete random variable \(X\) has \(E(X) = 4\) and \(E(X^2) = 25\).
Find the variance of \(X\), denoted as \(\text{Var}(X)\).

A discrete random variable \(X\) has a cumulative distribution function \(F(x) = P(X \le x)\) given by \(F(1) = 0.2\), \(F(2) = 0.6\), \(F(3) = 0.9\), and \(F(4) = 1.0\).
Find the probability \(P(X = 3)\).

A discrete random variable \(X\) has the probability distribution given by \(P(X = 1) = 0.2\), \(P(X = 2) = 0.5\), and \(P(X = 3) = 0.3\).
Find the expected value \(E(X)\).

A discrete random variable \(X\) has an expected value \(E(X) = 5\) and a variance \(\text{Var}(X) = 4\). If a new random variable \(Y\) is defined as \(Y = 10 - 2X\), calculate the value of \(E(Y^2)\).

The discrete random variable \(Y\) has the probability function:
\(P(Y=y) = c(y^2 + 1)\) for \(y = 1, 2, 3\).

(a) Show that \(c = \frac{1}{17}\).
(b) Find \(E(3Y - 2)\).
(c) Find \(Var(2Y + 5)\).

A box contains 3 red balls and 2 blue balls. Two balls are drawn at random without replacement. Let the random variable \(R\) be the number of red balls drawn.

(a) Construct the probability distribution table for \(R\).
(b) Find the cumulative distribution function \(F(r)\).
(c) Calculate the mean and variance of \(R\).