# area under curve normal distribution

#### billmt

##### New Member
Two issues, that I am confused about after reading statistics books.

Example:

weight of an adult is normally distributed with mean of 180 lbs and std dev of 20 lbs.

1) is the area under the curve equal to 100% or would the area be equal to the mean? The descriptions I see in statistics text books always indicate the probability but wouldn't the integral really be the mean (180 lbs.)?

2) what if I want to know the expected weight of an adult given the weight is above 200 lbs? all the examples I see only talk to what is the probability above or below 200 but I want to know the expected value given that it is above? what area of statistics addresses this question so I can read up on it? is there a formula for a conditional expectation for the normal curve?

Appreciate any help that any stats experts can provide.

#### ledzep

##### Point Mass at Zero
1) is the area under the curve equal to 100% or would the area be equal to the mean? The descriptions I see in statistics text books always indicate the probability but wouldn't the integral really be the mean (180 lbs.)?
Area under curve (AUC) is always 1 [or 100%].
Just to give you a motivating example, take standard normal curve which has mean 0 and variance=1.
If area under curve== mean (as you say), then the area under curve for a standard normal distribution will be zero. Does that makes sense?

2) what if I want to know the expected weight of an adult given the weight is above 200 lbs? all the examples I see only talk to what is the probability above or below 200 but I want to know the expected value given that it is above? what area of statistics addresses this question so I can read up on it? is there a formula for a conditional expectation for the normal curve?
You can use complementary probabilty here. If you have worked probabilty <=200 (call it p_a ), then probability >200 is 1-p_a.

HTH

#### billmt

##### New Member
thanks, this was really helpful . . . found exactly what I need