Suppose that X ~ N(μ,σ). Find a in terms of μ and σ if P(X>a) = 1/3 * P(X ≤a)
The Attempt at a Solution
1 - P(X ≤a) = 1/3 * P(X ≤a)
3 = 4P(X ≤a)
P(X ≤a) = 3/4
Since x0 = μ + σz0 where x0 and z0 are the same percentile for N(μ,σ) and N(0,1) (respectively), then z0 = 0.67449 (by invNorm(0.75, 0, 1)). It follows that
a = μ + 0.67449σ
I'm not sure if this is the correct method, someone in my class solved it another way and got a different answer, but this seems to make sense to me.