4. Let X equal the larger outcome when a pair of four-sided dice are rolled. Let D be the outcome space showing the paired outcomes of the two dice, with p = {(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)}. Assuming that the two dice are fair and independent, each of these outcomes has probability P({i, j}) = (a) Identify the outcome space of X, and its probability function f(x). (b) Find the mean of X. (c) The form of the probability function can be generalised to a pair of independent and fair m-sided dice, where X, the larger number shown, has probability function f(x) = 2x 1 m² x = 1,..., m. 2 Noting that ₁ = m(m+1)/2 and Σ² = m(m+1)(2m+1)/6, demon- strate that the probability function is valid, and confirm that the mean is (m+1)(4m-1) E(X) = 6m (d) Use the result in (c) to verify your answer in (b). NB: The first two parts are numerical. For the third part, you need not derive the formula for f(x), but rather check that it sums algebraically to the appropriate value. Use the definition of the mean of a discrete random variable, and seek to rearrange so that and can be exploited. The final part is computational.
4. Let X equal the larger outcome when a pair of four-sided dice are rolled. Let D be the outcome space showing the paired outcomes of the two dice, with p = {(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)}. Assuming that the two dice are fair and independent, each of these outcomes has probability P({i, j}) = (a) Identify the outcome space of X, and its probability function f(x). (b) Find the mean of X. (c) The form of the probability function can be generalised to a pair of independent and fair m-sided dice, where X, the larger number shown, has probability function f(x) = 2x 1 m² x = 1,..., m. 2 Noting that ₁ = m(m+1)/2 and Σ² = m(m+1)(2m+1)/6, demon- strate that the probability function is valid, and confirm that the mean is (m+1)(4m-1) E(X) = 6m (d) Use the result in (c) to verify your answer in (b). NB: The first two parts are numerical. For the third part, you need not derive the formula for f(x), but rather check that it sums algebraically to the appropriate value. Use the definition of the mean of a discrete random variable, and seek to rearrange so that and can be exploited. The final part is computational.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:4. Let X equal the larger outcome when a pair of four-sided dice are rolled. Let D
be the outcome space showing the paired outcomes of the two dice, with p =
{(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)}. Assuming that the two dice are fair and independent, each of
these outcomes has probability P({i, j})
=
(a) Identify the outcome space of X, and its probability function f(x).
(b) Find the mean of X.
(c) The form of the probability function can be generalised to a pair of independent
and fair m-sided dice, where X, the larger number shown, has probability
function
f(x) =
2x 1
m²
x = 1,..., m.
2
Noting that ₁ = m(m+1)/2 and Σ² = m(m+1)(2m+1)/6, demon-
strate that the probability function is valid, and confirm that the mean is
(m+1)(4m-1)
E(X) = 6m
(d) Use the result in (c) to verify your answer in (b).
NB: The first two parts are numerical. For the third part, you need not derive the formula
for f(x), but rather check that it sums algebraically to the appropriate value. Use the
definition of the mean of a discrete random variable, and seek to rearrange so that
and can be exploited. The final part is computational.
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