Question

Let e=(i,j) represent an arbitrary resulting from two rolls of the four-sided die of example 2.1.1. Tabulate the discrete pdf and sketch the graph of the CDF for the (a) Y(e) = i + j
(b) Z(e) = i - j
(c) W(e) = (i - j)^2

Answer 1

Answer on Question #79236 – Math – Statistics and Probability

Question

Let $e = (i,j)$ represent an arbitrary resulting from two rolls of the four-sided die of example 2.1.1. Tabulate the discrete pdf and sketch the graph of the CDF for the

(a) $Y(e) = i + j$

(b) $Z(e) = i - j$

(c) $W(e) = (i - j)^2$

Solution

The set of all possible outcomes is

$$(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2),$$

(4,3), (4,4). Each of the outcomes has probability 1/16.

a) $Y(e) = i + j$

The set of possible values is $\{2,3,4,5,6,7,8\}$. The probabilities are:

$$P(Y = 2) = P((1,1)) = 1/16$$ $$P(Y = 3) = P((1,2)) + P((2,1)) = 2/16$$ $$P(Y = 4) = P((1,3)) + P((2,2)) + P((3,1)) = 3/16$$ $$P(Y = 5) = P((1,4)) + P((2,3)) + P((3,2)) + P((4,1)) = 4/16$$ $$P(Y = 6) = P((2,4)) + P((3,3)) + P((4,2)) = 3/16$$ $$P(Y = 7) = P((3,4)) + P((4,3)) = 2/16$$ $$P(Y = 8) = P((4,4)) = 1/16$$

b) $Z(e) = i - j$

The set of possible values is $\{-3, -2, -1, 0, 1, 2, 3\}$

$$P(Z = -3) = P((1,4)) = 1/16$$ $$P(Z = -2) = P((1,3)) + P((2,4)) = 2/16$$ $$P(Z = -1) = P((1,2)) + P((2,3)) + P((3,4)) = 3/16$$ $$P(Z = 0) = P((1,1)) + P((2,2)) + P((3,3)) + P((4,4)) = 4/16$$ $$P(Z = 1) = P((2,1)) + P((3,2)) + P((4,3)) = 3/16$$ $$P(Z = 2) = P((3,1)) + P((4,2)) = 2/16$$ $$P(Z = 3) = P((4,1)) = 1/16$$

c) $W(e) = (i - j)^2$

The set of possible values is $\{0,1,4,9\}$

$$P(Z = 0) = P((1,1)) + P((2,2)) + P((3,3)) + P((4,4)) = 4/16$$ $$P(Z = 1) = P((1,2)) + P((2,3)) + P((3,4)) + P((2,1)) + P((3,2)) +$$ $$P((4,3)) = 6/16$$ $$P(Z = 4) = P((1,3)) + P((2,4)) + P((3,1)) + P((4,2)) = 4/16$$ $$P(Z = 9) = P((1,4)) + P((4,1)) = 2/16$$

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