# Normal Distribution Question

## Homework Statement

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.

Regards,

Michael

Homework Helper
Gold Member
2020 Award
I agree with your solution. Also please read my last comments on your Poisson distribution problem of last week. I think you'll find them of interest. That was actually the first time I had looked at the Poisson distribution in this much detail, and the two methods of solving the same problem are both quite interesting.

Of Mike and Men
Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

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.

Regards,

Michael

It is the correct method, and the answer is correct.