Question

A pair of dice is rolled. The random variable x is defined as the sum of the scores obtained. Find the probability function, mathematical expectation and variance

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Answer to a math question A pair of dice is rolled. The random variable x is defined as the sum of the scores obtained. Find the probability function, mathematical expectation and variance

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1. The possible sums $x$ that can be obtained when a pair of dice is rolled range from 2 to 12.
2. Calculate the probability function $P$X = x$$:

P$X = x$ = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

P$X = x$ = \begin{cases} \frac{x-1}{36} & x = 2,3,4,5,6,7 \\\frac{13-x}{36} & x = 8,9,10,11,12 \\0 & \text{otherwise}\end{cases}

3. Calculate the mathematical expectation $E$X$$:

E$X$ = \sum_{x=2}^{12} x \cdot P$X = x$

E$X$ = \sum_{x=2}^{7} x \cdot \frac{x-1}{36} + \sum_{x=8}^{12} x \cdot \frac{13-x}{36}

4. Simplify the summation to find $E$X$$:

E$X$ = \sum_{x=2}^{7} \frac{x$x-1$}{36} + \sum_{x=8}^{12} \frac{x$13-x$}{36}

E$X$ = 7

5. Calculate the variance $\text{Var}$X$$:

\text{Var}$X$ = E$X^2$ - [E$X$]^2

6. Calculate $E$X^2$$:

E$X^2$ = \sum_{x=2}^{12} x^2 \cdot P$X = x$

7. Simplify the summation to find $E$X^2$$ and then $\text{Var}$X$$:

E$X^2$ = \sum_{x=2}^{7} \frac{x^2$x-1$}{36} + \sum_{x=8}^{12} \frac{x^2$13-x$}{36}

\text{Var}$X$ = E$X^2$ - [E$X$]^2

\text{Var}$X$ = \frac{35}{6}

Therefore, the probability function, mathematical expectation, and variance are:

P$X = x$ = \begin{cases} \frac{x-1}{36} & x = 2,3,4,5,6,7 \\\frac{13-x}{36} & x = 8,9,10,11,12 \\0 & \text{otherwise}\end{cases}

E$X$ = 7

\text{Var}$X$ = \frac{35}{6}

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