Linear combinations of random variables：練習題

A random variable \(X\) has mean \(E(X) = 12\) and variance \(Var(X) = 5\). A new random variable \(Y\) is defined as \(Y = 3X - 4\). Calculate the mean and variance of \(Y\).

Two independent random variables \(X\) and \(Y\) have the following properties:
\(E(X) = 10, Var(X) = 4\)
\(E(Y) = 15, Var(Y) = 9\)

Find the standard deviation of the random variable \(W = 3X - 2Y + 7\).

The mass of an empty box, \(M_B\), follows a normal distribution \(N(100, 25)\) in grams. The mass of a single chocolate bar, \(C\), follows a normal distribution \(N(50, 16)\) in grams. A box is filled with 12 independent chocolate bars. Find the probability that the total mass of the full box exceeds \(720\) grams.

A random variable \(X\) has mean \(E(X) = 8\) and variance \(Var(X) = 2\). If a second random variable is defined as \(Y = 5 - 2X\), find the mean and variance of \(Y\).

Random variables $$X$$ and $$Y$$ are independent. Given that $$E(X) = 5$$, $$Var(X) = 2$$, $$E(Y) = 3$$, and $$Var(Y) = 4$$. Find $$E(2X - Y + 1)$$ and $$Var(2X - Y + 1)$$.

A random variable \(X\) has mean \(E(X) = 8\) and variance \(Var(X) = 2.5\). Find the mean and variance of the random variable \(Y = 4X + 7\).

Given two independent random variables, X and Y, with expectations and variances as follows: E(X) = 50, Var(X) = 9, E(Y) = 40, Var(Y) = 16. Calculate the expectation and variance of the linear combination \(2X - Y\).

Three independent random variables \(X_1, X_2, X_3\) are normally distributed with the following parameters:
\(X_1 \sim N(10, 2^2)\)
\(X_2 \sim N(12, 1.5^2)\)
\(X_3 \sim N(8, 2.5^2)\)
A new random variable S is defined as \(S = X_1 + X_2 + X_3\). Calculate \(P(S < 25)\).

A large bag contains a mixture of red and blue marbles. The weights, in grams, of individual red marbles are normally distributed with mean 15.4 and standard deviation 1.2. The weights, in grams, of individual blue marbles are normally distributed with mean 18.2 and standard deviation 1.5.

(a) Find the mean and variance of the total weight of 3 randomly chosen red marbles and 2 randomly chosen blue marbles.
(b) Calculate the probability that the total weight of these 5 marbles is less than 80 grams.

The random variable \(X\) has the distribution \(Po(2.5)\) and the random variable \(Y\) has the distribution \(Po(4.2)\). The variables \(X\) and \(Y\) are independent.

(a) State the distribution of \(X + Y\), including the value of its parameter.
(b) Find \(P(X + Y = 5)\).
(c) Given that \(W = 3X - 2\), find \(E(W)\) and \(Var(W)\).