Section 2.1

Random Variables

Question 2.1.1

Let \(\Omega = \{1,2,3,4\}\) and let \(X = I_{\{1,2\}}\), \(Y = I_{\{2,3\}}\), and \(Z = I_{\{3,4\}}\).

Let \(W = X + Y+ Z\).

1. Compute \(W(1)\).

NoteSolution to 2.1.1(a)

\[ \begin{aligned} W(1) &= X(1) + Y(1) + Z(1) \\ &= I_{\{1,2\}}(1) + I_{\{2,3\}}(1) + I_{\{3,4\}}(1) \\ &= 1 + 0 + 0\\ &= 1 \end{aligned} \]

2. Compute \(W(2)\).

NoteSolution to 2.1.1(b)

\[ \begin{aligned} W(2) &= X(1) + Y(2) + Z(2) \\ &= I_{\{1,2\}}(2) + I_{\{2,3\}}(2) + I_{\{3,4\}}(2) \\ &= 1 + 1 + 0\\ &= 2 \end{aligned} \]

Question 2.1.2

Let \(\Omega = \{1,2,3,4\}\) and \(X = I_{\{1,2\}}\) and \(Y(s) = s^2X(s)\).

1. Compute \(Y(1)\).

NoteSolution to 2.1.2(a)

\[ \begin{aligned} Y(2) &= (2)^2X(2)\\ &= (2)^2I_{\{1,2\}}(2)\\ &= 4(1)\\ &= 4 \end{aligned} \]

2. Compute \(Y(4)\).

NoteSolution to 2.1.2(b)

\[ \begin{aligned} Y(4) &= (4)^2X(4)\\ &= (4)^2I_{\{1,2\}}(4)\\ &= 16(0)\\ &= 0 \end{aligned} \]

Question 2.1.3

Suppose a student applies to enroll in two independent courses whose registrations are via a lottery. The student can be accepted (A), waitlisted (W), or rejected (R) into each course.

A students happiness is determined by the courses. For each course they are admitted to, their happiness increases by 50 points each. When waitlisted, their happiness increases by 15 points. When rejected, their happiness increases by 0 points.

1. Write out the sample space \(\Omega\).

NoteSolution to 2.1.2(b)

\(\Omega = \{AA, AW, WA, AR, RA, WW, RR, RW, WR\}\)

2. Define \(X\) as the total happiness score for student after applying to two courses. Enumerate how it operates for each \(\omega \in \Omega\).

NoteSolution to 2.1.2(b)

\[ X(\omega) = \begin{cases} 100 & \omega = AA\\ 65 & \omega = AW, WA\\ 50 & \omega = AR, RA\\ 30 &\omega = WW\\ 15 & \omega = WR, RW\\ 0 & \omega = RR \end{cases} \]