Probability distribution, expectation and variance：練習問題

Let \(X\) be a discrete random variable such that \(E[X] = 4\) and \(E[X^2] = 25\). Find the value of \(Var(3 - 2X)\).

A discrete random variable \( X \) has the probability distribution \( P(X=x) = \frac{x}{10} \) for \( x=1, 2, 3, 4 \). Find the variance of \( X^2 \).

The probability distribution of the number of siblings \(X\) in a certain community is shown in a bar chart. The values of \(X\) are \(0, 1, 2, 3\) with corresponding probabilities \(0.1, 0.4, 0.3,\) and \(0.2\). Calculate the expected number of siblings per household.

Suppose \( X \) is a discrete random variable such that \( E[X] = 10 \) and \( Var(X) = 4 \). Find the value of \( E[X^2 - 2X + 5] \).

Consider a discrete random variable \(Y\) such that \(P(Y=y) = \frac{c}{y!}\) for \(y = 0, 1, 2, 3\). It is given that the variance of \(Y\) is denoted by \(Var(Y)\).
(a) Determine the constant \(c\) in terms of a simplified fraction.
(b) Find the value of \(Var(2 - 3Y)\), correct to 4 decimal places.

A discrete random variable \(X\) has a probability distribution given by \(P(X=x) = \frac{kx}{2^x}\) for \(x = 1, 2, 3, 4\) and \(P(X=x) = 0\) otherwise.
Find the value of the constant \(k\) and evaluate \(Var(3 - 2X)\), leaving your answer as a fraction.

A discrete random variable \(X\) has the probability distribution given by \(P(X=x) = \frac{k}{x(x+1)}\) for \(x = 1, 2, 3, 4\).
(a) Find the value of the constant \(k\).
(b) Evaluate the expectation \(E[5X - 2]\).

A discrete random variable \(X\) has the probability distribution as follows:
\(P(X = -1) = k\)
\(P(X = 0) = 2k\)
\(P(X = 1) = 1 - 4k\)
\(P(X = 2) = k\)
where \(k\) is a constant.

(a) Find the range of possible values of \(k\).
(b) Given that \(E[X^2] = 1.2\), find the value of \(k\).
(c) Find the variance of \(2 - 3X\).
(d) Find \(E[X^3 - X]\).

A discrete random variable $$X$$ has a probability distribution given by $$P(X=x) = k(x^2+1)$$ for $$x=1, 2, 3$$, and $$P(X=x)=0$$ otherwise.

a) Find the value of the constant $$k$$.

b) Construct the probability distribution table for $$X$$.

c) Find $$E(X)$$.

d) Find $$Var(X)$$.

e) Let $$Y = 2X - 3$$. Find $$E(Y)$$ and $$Var(Y)$$.