Problems from Average, variance and standard deviation

We have the following discrete random: If the result of throwing a perfect dice is a prime number, the payoff will be the result times $$10$$. We include in the table these payoffs. Assign payoffs to the other results from throwing the dice.

Result of the dice probability payoff
$$1$$ $$1/6$$ $$10$$
$$2$$ ? ?
$$3$$ ? $$30$$
$$4$$ ? ?
$$5$$ $$1/6$$ ?
$$6$$ $$1/6$$ ?
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Development:

Result of the dice probability payoff
$$1$$ $$1/6$$ $$10$$
$$2$$ $$1/6$$ $$20$$
$$3$$ $$1/6$$ $$30$$
$$4$$ $$1/6$$ $$8$$
$$5$$ $$1/6$$ $$50$$
$$6$$ $$1/6$$ $$120$$

  • $$$\mu=\sum_i p_i\cdot x_i=\dfrac{1}{6}\cdot10+\dfrac{1}{6}\cdot20+\dfrac{1}{6}\cdot30+\dfrac{1}{6}\cdot8+\dfrac{1}{6}\cdot50+\dfrac{1}{6}\cdot120$$$ $$$\mu=\dfrac{238}{6}=39,67$$$

  • The variance is calculated first: $$$\sigma^2=\sum_i x_i^2\cdot p_i - \mu^2=\dfrac{1}{6}(10^2+20^2+30^2+8^2+50^2+120^2)-39,67^2$$$

    variance $$\rightarrow \sigma^2=1486,95$$

    standard deviation $$\rightarrow \sigma=38,56$$

Solution:

Result of the dice probability payoff
$$1$$ $$1/6$$ $$10$$
$$2$$ $$1/6$$ $$20$$
$$3$$ $$1/6$$ $$30$$
$$4$$ $$1/6$$ $$8$$
$$5$$ $$1/6$$ $$50$$
$$6$$ $$1/6$$ $$120$$

  • $$\mu=\dfrac{238}{6}=39,67$$

  • variance $$\rightarrow \sigma^2=1486,95$$

    standard deviation $$\rightarrow \sigma=38,56$$

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