Chapter 7: Q.7.76 (page 358)

Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled. Compute the joint moment generating function of $X$ and $Y$ .

Short Answer

Expert verified

The joint moment generating functions of $X$ and $Y$ are $M_{x, y} (t_{1}, t_{2}) = \frac{1}{36} \sum_{x = 1}^{6} \sum_{y = x + 1}^{x + 6} e^{t_{1} x + t_{2} y}$ .

Step by step solution

Given Information

Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled.

Explanation

$Y = \{\begin{cases} 2, 3, 4, 5, 6, 7 i f X = 1 \\ 3, 4, 5, 6, 7, 8 i f X = 2 \\ 4, 5, 6, 7, 8, 9 i f X = 3 \\ 5, 6, 7, 8, 9, 10 i f X = 4 \\ 6, 7, 8, 9, 10, 11 i f X = 5 \\ 7, 8, 9, 10, 11, 12 i f X = 6 \end{cases}$

$P (X = x, Y = y) = \frac{1}{36}$ for all $x$ and $y$ .

M_{x, y} (t_{1}, t_{2}) = E [e^{t_{1} X + t_{2} Y}]

Explanation

$M_{x, y} (t_{1}, t_{2}) = E [e^{t_{1} X + t_{2} Y}]$

= \sum_{x = 1}^{6} \sum_{y = x + 1}^{x + 6} P (X = x, Y = y) e^{t_{1} x + t_{2} y}

$= \frac{1}{36} \sum_{x = 1}^{6} \sum_{y = x + 1}^{x + 6} e^{t_{1} x + t_{2} y}$

Final Answer

The joint moment generating functions of $X$ and $Y$ are $M_{x, y} (t_{1}, t_{2}) = \frac{1}{36} \sum_{x = 1}^{6} \sum_{y = x + 1}^{x + 6} e^{t_{1} x + t_{2} y}$ .

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Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled. Compute the joint moment generating function of $X$ and $Y$ .

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final Answer

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Let X be the value of the first die and Ythe sum of the values when two dice are rolled. Compute the joint moment generating function of X and Y.

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Pure Maths

Mechanics Maths

Geometry

Study anywhere. Anytime. Across all devices.

Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled. Compute the joint moment generating function of $X$ and $Y$ .