The amount of money that Jon can save after working for a summer is a random variable S with a mean of $μs\mu _ { \mathrm { s } }$ and a standard deviation of $σs\sigma _ { s }$ .After saving this money Jon plans to go on a trip to India.He will change his money into Rupees at an exchange rate of 43 Rupees to one Dollar.This money he will bring to India.When he arrives in India he will buy a used motorbike.The price in India of a motorbike of the type he wants is a random variable B with a mean of $μb\mu \mathrm { b }$ Rupees and a standard deviation of $σb\sigma b$ Rupees. The amount of money Jon will have left (in Rupees) after changing his savings into Rupees and buying a motorbike in India is a random variable P which can be expressed in terms of S and B as $P = 43 S - B$ Find expressions for the mean and variance of the random variable P.Assume that Jon's savings and the price of the bike are independent.

A) mean = 43 $μs\mu _ { \mathrm { s } }$ - $μb\mu \mathrm { b }$ ,variance = 1849 $σs2\sigma _ { s } ^ { 2 }$ - $σb2\sigma _ { b } ^ { 2 }$
B) mean = 43 $μs\mu _ { \mathrm { s } }$ - $μb\mu \mathrm { b }$ ,variance = 1849 $σs2\sigma _ { s } ^ { 2 }$ + $σb2\sigma _ { b } ^ { 2 }$
C) mean = 43 $μs\mu _ { \mathrm { s } }$ + $μb\mu \mathrm { b }$ ,variance = 1849 $σs2\sigma _ { s } ^ { 2 }$ + $σb2\sigma _ { b } ^ { 2 }$
D) mean = 43 $μs\mu _ { \mathrm { s } }$ + $μb\mu \mathrm { b }$ ,variance = 43 $σs2\sigma _ { s } ^ { 2 }$ + $σb2\sigma _ { b } ^ { 2 }$
E) mean = 43 $μs\mu _ { \mathrm { s } }$ - $μb\mu \mathrm { b }$ ,variance = 43 $σs2\sigma _ { s } ^ { 2 }$ + $σb2\sigma _ { b } ^ { 2 }$

Question

The amount of money that Jon can save after working for a summer is a random variable S with a mean of

μs\mu _ { \mathrm { s } }

and a standard deviation of

σs\sigma _ { s }

.After saving this money Jon plans to go on a trip to India.He will change his money into Rupees at an exchange rate of 43 Rupees to one Dollar.This money he will bring to India.When he arrives in India he will buy a used motorbike.The price in India of a motorbike of the type he wants is a random variable B with a mean of

μb\mu \mathrm { b }

Rupees and a standard deviation of

σb\sigma b

Rupees. The amount of money Jon will have left (in Rupees) after changing his savings into Rupees and buying a motorbike in India is a random variable P which can be expressed in terms of S and B as

P = 43 S - B

Find expressions for the mean and variance of the random variable P.Assume that Jon's savings and the price of the bike are independent.

A) mean = 43

μs\mu _ { \mathrm { s } }

-

μb\mu \mathrm { b }

,variance = 1849

σs2\sigma _ { s } ^ { 2 }

-

σb2\sigma _ { b } ^ { 2 }

B) mean = 43

μs\mu _ { \mathrm { s } }

-

μb\mu \mathrm { b }

,variance = 1849

σs2\sigma _ { s } ^ { 2 }

+

σb2\sigma _ { b } ^ { 2 }

C) mean = 43

μs\mu _ { \mathrm { s } }

+

μb\mu \mathrm { b }

,variance = 1849

σs2\sigma _ { s } ^ { 2 }

+

σb2\sigma _ { b } ^ { 2 }

D) mean = 43

μs\mu _ { \mathrm { s } }

+

μb\mu \mathrm { b }

,variance = 43

σs2\sigma _ { s } ^ { 2 }

+

σb2\sigma _ { b } ^ { 2 }

E) mean = 43

μs\mu _ { \mathrm { s } }

-

μb\mu \mathrm { b }

,variance = 43

σs2\sigma _ { s } ^ { 2 }

+

σb2\sigma _ { b } ^ { 2 }

Aibiike Esengulova · Accepted Answer

Final Answer : B, Explanation :   The mean of P is the mean of 43S minus the mean of B, which is   43 μ s − μ b 43\mu_s - \mu_b 43 μ s ​ − μ b ​  . The variance of P is the variance of 43S plus the variance of B, since S and B are independent, which is   1849 σ s 2 + σ b 2 1849\sigma_s^2 + \sigma_b^2 1849 σ s 2 ​ + σ b 2 ​   (because variance of a constant times a variable is the constant squared times the variance of the variable).

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