Question$2 X \sim N\left(\mu, 15^{2}\right)$. Given that $P(X<90)=0.1$, determine $\mu$. $3 X \sim N\left(40, \sigma^{2}\right)$. Given that $P(X>50)=0.01$, determine $\mu$. $4 X \sim N\left(\mu, \sigma^{2}\right)$. Given that $P(X<5)=0.05$ and $P(X>10)=0.01$, determine $\mu$ and $\sigma$.

Studdy Solution
Subtract the first equation from the second:
$$2.3263 + 1.6449 = \frac{10 - 5}{\sigma}$$
$$\sigma = \frac{5}{3.9712}$$
$$\sigma \approx 1.259$$
Substitute $\sigma$ back into one of the equations to find $\mu$:
$$-1.6449 = \frac{5 - \mu}{1.259}$$
$$\mu = 5 + 1.6449 \times 1.259$$
$$\mu \approx 6.07$$
The solutions are:
1. For $2X$, $\mu \approx 199.224$.
2. For $3X$, $\sigma \approx 47.32$.
3. For $4X$, $\mu \approx 6.07$ and $\sigma \approx 1.259$.