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1. Normal density at a point ●○○○○ 10 ptsFind $P(x)$ for the normal distribution with mean $\mu = 2$ and standard deviation $\sigma = 3$ at $x = 5$, using $$P(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\,e^{-(x-\mu)^2/(2\sigma^2)}.$$ Use $\pi \approx 3.14$. Round to four decimal places.
2. Outlier percentage in a normal distribution ●●○○○ 20 ptsIn a normal distribution, a value is an outlier if it lies more than $1.5$ interquartile ranges above the upper quartile or more than $1.5$ interquartile ranges below the lower quartile. Using $z=\pm 0.67$ for the quartiles, what percent of values are outliers under this definition? Give the percent rounded to two decimal places.
3. Mean of a normal distribution ●●○○○ 20 ptsLet $X$ be normally distributed. Given that $P(-3<X<-1)=P(3<X<5)$, find the expectation $E(X)$.