Independent solution

How to solve this Normal Distribution question

Setup

Setup

Write the exponential distribution through its survival function and solve for its 80th percentile q.

0.20=e−q/5000.20=e^{-q/500}
q=500log⁡5=804.7189562…q=500\log 5=804.7189562\ldots

Model

Model

For the normal model, the same percentile is the mean plus the 80th standard-normal quantile times the unknown standard deviation.

q=500+z0.80σq=500+z_{0.80}\sigma
z0.80=0.8416212336…z_{0.80}=0.8416212336\ldots

Compute

Compute

Solve for the normal standard deviation, then use the identity relating the raw second moment to variance and squared mean.

σ=500log⁡5−5000.8416212336=362.0618683…\sigma=\frac{500\log 5-500}{0.8416212336}=362.0618683\ldots
E⁡[X2]=σ2+μ2=(362.0618683)2+5002\operatorname{E}[X^2]=\sigma^2+\mu^2=(362.0618683)^2+500^2
E⁡[X2]=381088.7965…\operatorname{E}[X^2]=381088.7965\ldots

Answer

Answer

Rounding the second moment to the nearest thousand gives 381,000.

381,000(C)\boxed{381{,}000\quad\text{(C)}}