## Related questions with answers

If X is normal with mean 12 and standard deviation 2 , find $P(10<X \leq 15)$.

Solution

VerifiedWe are given the normal random variable $X$ whose mean has value $\mu=12$ and standard deviation has value $\sigma=2$. Goal is to calculate the probability $P(10<X\leq15)$. Let's try to standardize this probability. For any normal random variable with mean $\mu$ and standard deviation $\sigma$, the following equality is true:

$\begin{aligned} % % Remove numbering (before each equation) P(c_1<X<c_2) &=& P\left(\frac{c_1-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{c_2-\mu}{\sigma}\right) \\ &=& P\left(\frac{c_1-\mu}{\sigma}<Z<\frac{c_2-\mu}{\sigma}\right) \end{aligned}$

Since we have values for mean and standard deviation, we get that:

$\begin{aligned} % % Remove numbering (before each equation) \textcolor{#c34632}{P(10<X\leq15)} &=& P\left(\frac{10-12}{2}<\frac{X-0.05}{0.01}\leq\frac{15-12}{2}\right) \\ &=& P\left(\frac{-2}{2}<Z\leq\frac{3}{2}\right) \\ &=& \textcolor{#c34632}{P\left(-1<Z\leq1.5\right)} \end{aligned}$

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