In a game played with dice, Ali or Bob is selected with a probability of 5 and , respectively. The dice are fair and four-sided with the values of 1, 2, 3 and 4. If selected, Ali will roll a die until he obtains the second value of 4 and the number of rolls required is recorded. If selected, Bob will roll two dice and the sum of the obtained values is recorded. Let X denote the recorded outcome of the game. Find P(X ≤ 3). Show that the moment generating function (mgf) for X is 2 et 3³ (4-130²) + 5 3et 40 (i) (ii) (iii) M (t) = 4 x=1 Hence, using part (ii), find the mean of X. 2 t < ln 4
In a game played with dice, Ali or Bob is selected with a probability of 5 and , respectively. The dice are fair and four-sided with the values of 1, 2, 3 and 4. If selected, Ali will roll a die until he obtains the second value of 4 and the number of rolls required is recorded. If selected, Bob will roll two dice and the sum of the obtained values is recorded. Let X denote the recorded outcome of the game. Find P(X ≤ 3). Show that the moment generating function (mgf) for X is 2 et 3³ (4-130²) + 5 3et 40 (i) (ii) (iii) M (t) = 4 x=1 Hence, using part (ii), find the mean of X. 2 t < ln 4
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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In a game played with dice, Ali or Bob is selected with a probability of
respectively. The dice are fair and four-sided with the values of 1,
2, 3 and 4. If selected, Ali will roll a die until he obtains the second value
of 4 and the number of rolls required is recorded. If selected, Bob will roll
two dice and the sum of the obtained values is recorded. Let X denote
the recorded outcome of the game.
Find P(X ≤ 3).
Show that the moment generating function (mgf) for X is
2
et
M(C) = ² ( = = 12 ) ² + + (20²)² = + < In (1)
1
4
Σ
ptx
4 3et 40
(i)
(ii)
(iii)
x=1
315
Hence, using part (ii), find the mean of X.
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