Question

The standard deviation of a normal random variable x is $20. given that p(x ≤ $10) = 0.1841. from this we can determine that the mean of the distribution is equal to

Answer 1

Let $\mu$ be the mean of $X$. Then

$\mathbb P(X\le10)=0.1841\iff\mathbb P\left(\underbrace{\dfrac{X-\mu}{20}}_Z\le\dfrac{10-\mu}{20}\right)=0.1841$

For a random variable $Z$ following the standard normal distribution, we have

$\mathbb P(Z\le z)=0.1841\implies z\approx-0.8999$

Transform the random variable to get this critical value in terms of $\mu$:

$\dfrac{10-\mu}{20}=-0.8999\implies\mu\approx27.997$