Normal distribution help?

Question

Random vairable X is normally distributed with mean u and variance 02. Given that P(X>58.37)=0.02 and P(X<40.85)=0.01, calculate u and o2..

Answer 1

You have to first get a table of areas under the normal curve.

My table gives areas from the mean out to any value of z, where z is a standard normal variate (mean 0, sd 1)

Next, P(X>58.37)=0.02 means the area under the curve (for the table I have) should be 0.48 and my table shows this happens at z=2.055. So (58.37-u)/o = 2.055.

Next, P(x<40.85)=0.01 means the area under the curve (for my table) is 0.49 & this happens at z=2.33. So
(40.85-u)/o=2.33.

You now have 2 equations in 2 unknowns, & they can be solved for u & o. Square o to o2.

Answer 2

First you normalize your Gaussian curve using substitution Y = (X - u)/s. Then you look at the table of the probability integral. There you find:

(58.37 - u)/s = 2.055, and
(40.85 - u)/s = -2.325,

where s = standard deviation (sigma): s = sqrt(o2).

From there you get: s = 4, o2 = s^2 = 16, and u = 50.