An auto dealer offers discounts averaging 2450 with a...

An auto dealer offers discounts averaging $$\$ 2450$$ with a standard deviation of $$\$ 575 .$$ If 50 autos are sold during one month, what is the expected value and standard deviation for the total discounts given? (A) $E($ Total $)=\$ 17,324 \quad \mathrm{SD}($ Total $)=\$ 170$ (B) $E($ Total $)=\$ 17,324 \quad \mathrm{SD}($ Total $)=\$ 4066$ (C) $E($ Total $)=\$ 122,500 \quad \mathrm{SD}($ Total $)=\$ 170$ (E) $E($ Total $)=\$ 122,500 \quad \mathrm{SD}($ Total $)=\$ 28,750$

An auto dealer offers discounts averaging $$\$ 2450$$ with a standard deviation of $$\$ 575 .$$ If 50 autos are sold during one month, what is the expected value and standard deviation for the total discounts given?
(A) $E($ Total $)=\$ 17,324 \quad \mathrm{SD}($ Total $)=\$ 170$
(B) $E($ Total $)=\$ 17,324 \quad \mathrm{SD}($ Total $)=\$ 4066$
(C) $E($ Total $)=\$ 122,500 \quad \mathrm{SD}($ Total $)=\$ 170$
(E) $E($ Total $)=\$ 122,500 \quad \mathrm{SD}($ Total $)=\$ 28,750$

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Key Concepts

Linearity of Expectation

This principle states that the expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether the variables are independent. In practical terms, if each transaction has an average discount, the total expected discount is simply the number of transactions multiplied by the average discount.

Variance and Standard Deviation for Independent Sums

For independent random variables, the variance of their sum is equal to the sum of their individual variances. Since the standard deviation is the square root of the variance, the standard deviation of the sum is the square root of the number of variables multiplied by the individual standard deviation, provided all have the same variance. This concept is key in calculating the spread of the total discount from many sales.

Central Limit Theorem

When summing a large number of random variables, regardless of the distribution of each individual variable, the distribution of the sum tends towards a normal distribution. This principle justifies using mean and standard deviation to describe the total discount's distribution in scenarios like large sales volumes.

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