Continuous random variables：練習問題

A continuous random variable \( X \) has the probability density function \( f(x) = kx^2(1-x) \) for \( 0 \le x \le 1 \) and \( 0 \) otherwise. Find the value of the constant \( k \) and the variance \( \text{Var}(X) \).

The continuous random variable \( X \) has the probability density function \( f(x) = \frac{1}{2}e^{-|x|} \) for all real \( x \). The random variable \( Y \) is defined by \( Y = e^{|X|} \). Find the probability density function \( g(y) \) of \( Y \) for \( y \ge 1 \).

A continuous random variable \( X \) has a probability density function defined by:
\( f(x) = \begin{cases} \frac{3}{16}(4x - x^2) & 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases} \)
Find the value of the constant \( a \) such that \( P(X > a) = \frac{11}{32} \).

The continuous random variable \( X \) has probability density function \( f(x) = \frac{1}{2}x \) for \( 0 \le x \le 2 \). The random variable \( Y \) is defined by \( Y = X^3 \). Find the probability density function \( g(y) \) for \( 0 \le y \le 8 \).

The continuous random variable \( X \) has a cumulative distribution function \( F(x) \) given by:
\( F(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{8}x^3 & 0 \le x \le 2 \\ 1 & x > 2 \end{cases} \)
Find the value of \( E(X^2) \).

The continuous random variable \(X\) has the probability density function \(f(x) = k(x-1)(3-x)\) for \(1 \le x \le 3\) and zero otherwise. Find the value of \(k\) and show that the variance of \(X\) is \(0.2\).