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Question 3: You are ordering CPUs from a chip manufacturer, the same as Question 2 from the previous assignment. As in the last assignment, the chip manufacturer manufactures 1000 CPUs. 75 of the CPUs are defective, and are distributed uniformly at random throughout the 1000 CPUs. You purchase 150 CPUs from the company. Let X be the number of defective CPUs you purchase. You are going to find E(X) using three different methods. a) Determine E(X) using the definition of expected value, that is 4 E(X) = \sum_{k=0}^{4} k \cdot Pr(X = k). Since this expression will be difficult to simplify, you may leave it as a summation. Or you can use Wolfram Alpha to get the solution. b) We will give each defective CPUs a number 1 through 75. Define indicator random variables A1, A2, A3, ..., A75 such that for i ? {1..75}: \{ \begin{array}{rcl} A_i &=& 1 \\ A_i &=& 0 \end{array} \begin{array}{rcl} \text{if you purchased defective CPU number } i, \text{and} \\ \text{otherwise.} \end{array} Determine E(X) using these indicator random variables. c) Define indicator random variables C1, C2, C3, ..., C150 representing the CPUs that you purchased. For i ? {1..150}, let: \{ \begin{array}{rcl} C_i &=& 1 \\ C_i &=& 0 \end{array} \begin{array}{rcl} \text{if CPU } i \text{ is defective} \\ \text{otherwise.} \end{array} Determine E(X) using these indicator random variables.

          Question 3: You are ordering CPUs from a chip manufacturer, the same as Question 2 from
the previous assignment. As in the last assignment, the chip manufacturer manufactures 1000
CPUs. 75 of the CPUs are defective, and are distributed uniformly at random throughout
the 1000 CPUs. You purchase 150 CPUs from the company. Let X be the number of
defective CPUs you purchase. You are going to find E(X) using three different methods.
a) Determine E(X) using the definition of expected value, that is
4
E(X) = \sum_{k=0}^{4} k \cdot Pr(X = k).
Since this expression will be difficult to simplify, you may leave it as a summation. Or
you can use Wolfram Alpha to get the solution.
b) We will give each defective CPUs a number 1 through 75. Define indicator random
variables A1, A2, A3, ..., A75 such that for i ? {1..75}:
\{ \begin{array}{rcl} A_i &=& 1 \\ A_i &=& 0 \end{array} \begin{array}{rcl} \text{if you purchased defective CPU number } i, \text{and} \\ \text{otherwise.} \end{array}
Determine E(X) using these indicator random variables.
c) Define indicator random variables C1, C2, C3, ..., C150 representing the CPUs that you
purchased. For i ? {1..150}, let:
\{ \begin{array}{rcl} C_i &=& 1 \\ C_i &=& 0 \end{array} \begin{array}{rcl} \text{if CPU } i \text{ is defective} \\ \text{otherwise.} \end{array}
Determine E(X) using these indicator random variables.

$Question 3: You are ordering CPUs from a chip manufacturer, the same as Question 2 from the previous assignment. As in the last assignment, the chip manufacturer manufactures 1000 CPUs. 75 of the CPUs are defective, and are distributed uniformly at random throughout the 1000 CPUs. You purchase 150 CPUs from the company. Let X be the number of defective CPUs you purchase. You are going to find E(X) using three different methods. a) Determine E(X) using the definition of expected value, that is 4 E(X) = ∑k=0^4 k ·Pr(X = k). Since this expression will be difficult to simplify, you may leave it as a summation. Or you can use Wolfram Alpha to get the solution. b) We will give each defective CPUs a number 1 through 75. Define indicator random variables A1, A2, A3, ..., A75 such that for i ? 1..75: { Ai = 1 Ai = 0 if you purchased defective CPU number i, and otherwise. Determine E(X) using these indicator random variables. c) Define indicator random variables C1, C2, C3, ..., C150 representing the CPUs that you purchased. For i ? 1..150, let: { Ci = 1 Ci = 0 if CPU i is defective otherwise. Determine E(X) using these indicator random variables.$

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