Discrete random variables：練習問題

The discrete random variable \(X\) has expected value \(E(X) = 1.5\) and variance \(Var(X) = 0.75\). Calculate the value of \(E(2X^2)\).

The probability function of a discrete random variable \(X\) is given by \(P(X=x) = k(3-x)\) for \(x = 0, 1, 2\). Find the value of \(Var(3X+1)\).

The discrete random variable \(X\) has expected value \(E(X) = 3\) and \(E(X^2) = 15\). Find the value of \(E((2X - 1)^2)\).

Let \(X\) and \(Y\) be independent discrete random variables with \(Var(X) = 5\) and \(Var(Y) = 8\). A new random variable \(W\) is defined such that \(W = 4X - 2Y + 7\). Find the variance of \(W\).

The discrete random variable \(Y\) has the following probability distribution: \(P(Y=1) = 0.2\), \(P(Y=2) = 0.3\), and \(P(Y=5) = 0.5\). Find the value of \(E(Y^2)\).

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

The discrete random variable \(X\) takes values 1, 2, and 3 only. Given that \(P(X=1) = p\), \(P(X=3) = p\), and \(E(X) = 2\), show that \(Var(X) = 2p\).

The discrete random variable \(X\) has mean \(E(X) = 5\) and variance \(Var(X) = 2\). Find the value of \(E(X^2 + 2X)\).

A discrete random variable \(X\) has the probability distribution given by the following table:

x: 1, 2, 3, 4, 5
P(X = x): \(k\), \(2k\), \(3k\), \(4k\), \(5k\)

(a) Show that the value of the constant \(k\) is \(\frac{1}{15}\).
(b) Calculate the expected value \(E(X)\).
(c) Determine the value of \(P(X > E(X))\).

The random variable \(X\) has the following probability distribution:

x: 1, 2, 3, 4
P(X = x): \(c(5 - x)\)

(a) Find the exact value of \(c\).
(b) Calculate the mean, \(\mu\).
(c) Show that \(Var(X) = 1\).