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.

• Fill in the following table:
 Result of the dice probability payoff $$1$$ $$1/6$$ $$10$$ $$2$$ ? ? $$3$$ ? $$30$$ $$4$$ ? ? $$5$$ $$1/6$$ ? $$6$$ $$1/6$$ ?
• Find the average payoff if we throw the dice only once.

• Find the variance and the standard deviation.
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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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