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May 2nd, 2017, 05:18 AM   #1
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Stuck on probability distribution of a normal distribution problem

https://gyazo.com/921c2e7fd4ace575e11c5fc0afcedee0
Any ideas on how to approach part iii? Thanks alot
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May 2nd, 2017, 10:51 AM   #2
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you should know that if two normal random variables

$X \sim N(\mu_x, \sigma_x),~Y \sim N(\mu_y, \sigma_y)$

then

$X + Y \sim N\left(\mu_x+\mu_y,\sqrt{\sigma_x^2 + \sigma_y^2}\right)$

and

$c X \sim N(c \mu_x, c \sigma_x)$

so the mean of $N$ identically distributed normal random variables, $X$, will be distributed as

$\bar{X} \sim N\left(\mu_x, \dfrac{\sigma_x}{\sqrt{N}}\right)$

this should let you complete part 3
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