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November 13th, 2017, 11:35 AM   #1
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Question Find standard deviation given normal probability

I have

$X \sim N(0,\sigma^2), Y \sim N(-1,4) \mbox{ independent}, \\ Z = X-Y$

find $\sigma$ such that $P(Z \le 7) = 0.84134$

what I have done is :

$X-Y \sim N(0-(-1), \sigma^2+4) = (1, \sigma^2+4)$

$\begin{array}{lcl}P(Z \le 7) & = & P( \frac{Z -1}{\sqrt{\sigma^2+4}} \le \frac{7-1}{\sqrt{\sigma^2+4}}) \\ & = & P(W \le \frac{6}{\sqrt{\sigma^2+4}}) &= & 0.84137 \end{array}$

$W \sim N(0,1) \\ 0.84137 = \phi(1) \mbox{ therefore }$

$\begin{array}{lcl}\frac{6}{\sqrt{\sigma^2+4}} & = & 1 \\ \frac{36}{\sigma^2+4} & = & 1 \\ \frac{36(\sigma^2+4)}{\sigma^2+4} & = & \sigma^2+4 \\ 36 & = & \sigma^2 + 4 \\ \sigma^2 &= & 32 \\ \sigma & = & \sqrt{32} \end{array}$

please, can you tell me if it's correct? Thanks!
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November 14th, 2017, 01:51 AM   #2
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seems correct
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