離散隨機變量：練習題

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

\(P(X=0) = 0.4\)
\(P(X=1) = 0.3\)
\(P(X=2) = 0.2\)
\(P(X=3) = 0.1\)

Calculate the expected value \(E(X)\).

A discrete random variable \( X \) has a probability distribution defined by the function:
\( P(X = x) = \begin{cases} kx^2 & x = 1, 2, 3 \\ 0 & \text{otherwise} \end{cases} \)
Find the value of the constant \( k \) and use it to calculate the expected value \( E(X) \).

A discrete random variable \(X\) can take values 1, 2, and 3. Given that \(P(X=1) = 0.4\) and \(P(X=2) = 0.3\), calculate the expected value \(E(X)\).

A discrete random variable \(X\) has the probability distribution function defined by \(P(X=x) = \frac{x^2}{k}\) for \(x = 1, 2, 3, 4\). Calculate the variance of \(X\).

A continuous random variable \(X\) follows a continuous uniform distribution over the interval \([a, b]\). The probability density function is given by \(f(x) = k\) for \(a \le x \le b\) and \(0\) otherwise.

Given that \(E(X) = 7\) and \(Var(X) = 3\), determine the values of \(a\) and \(b\).

A discrete random variable \(X\) has the following probability distribution:
x | 0 | 1 | 2 | 3 | 4
P(X=x) | 0.1 | 0.2 | 0.4 | 0.2 | 0.1
(a) Calculate the expected value \(E(X)\). (2 points)
(b) Calculate the variance \(Var(X)\). (3 points)
(c) Two independent observations of \(X\), denoted \(X_1\) and \(X_2\), are taken. Find the probability that \(X_1 + X_2 = 2\). (3 points)< /p>

A discrete random variable \(X\) has a probability distribution defined by the function:
\(P(X = x) = \begin{cases} k(x+2)^2 & x = -1, 0, 1 \\ c & x = 2 \\ 0 & \text{otherwise} \end{cases}\)
where \(k\) and \(c\) are constants.

Given that the expected value \(E(X) = 0.9\), determine:
(a) The values of the constants \(k\) and \(c\). (3 points)
(b) The exact value of \(Var(X)\). (3 points)
(c) The standard deviation of the linear transformation \(W = 4 - 5X\). (2 points)