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{(X-\mu)/(20)}_Z\le(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`:


`(10-\mu)/(20)=-0.8999\implies\mu\approx27.997`