Discrete random variables：練習問題

A discrete random variable \(X\) has a probability distribution such that \(P(X=0) = 0.35\) and \(P(X=1) = 0.45\). If \(X\) can only take values \(0, 1,\) and \(2\), find the value of \(P(X=2)\).

A discrete random variable \(X\) has possible values \(-2, -1, 0, 1, 2\). Its probability distribution is given by \(P(X=x) = \frac{|x|+a}{10}\). Find the value of the constant \(a\), and then calculate \(E[X]\).

A game involves rolling a fair four-sided die with faces numbered 1, 2, 3, 4. If the die shows an odd number, you win the number of dollars shown. If the die shows an even number, you lose 2 dollars. Let \(Y\) be the random variable representing your net winnings in a single roll. Calculate \(E[Y^2]\).

A discrete random variable \( X \) has a probability distribution given by \( P(X=x) = k(x+1) \) for \( x=0, 1, 2, 3 \). Find the value of the constant \( k \).

The discrete random variable \(X\) has a probability distribution given by \(P(X=x) = \frac{c}{x!}\) for \(x=0, 1, 2, 3\). Find the value of the constant \(c\), and then determine \(Var(2X-1)\).

A fair coin is tossed 3 times. Let \(X\) be the random variable representing the number of heads obtained. Briefly explain why \(X\) is classified as a discrete random variable.

An urn contains 3 red balls and \(n\) blue balls. Two balls are drawn at random without replacement. Let \(B\) be the event that the first ball is blue and \(R\) be the event that the second ball is red. If the conditional probability \(P(R|B) = \frac{1}{3}\), find the value of \(n\) and determine if events \(B\) and \(R\) are independent.