Normal Probabilities
For several values of z the tables I provided in class (and that are available
here) give you P(Z > z). For
instance, P(Z > 1.96) = 0.0250
Since the normal curve is symmetric, it is also easy to get probabilities from the
other side of the pdf directly from the table. P(Z > 1.96) =
P(Z < -1.96) = 0.0250.
Also since the area under the normal curve is 1, P(Z < 1.96) =
1 - P(Z > 1.96) = 1 - 0.0250 = 0.9750
Since the normal pdf is symmetric around zero, P(Z > 0) =
P(Z < 0) = 0.5. We can use this fact to find probabilities such as
P(0 < Z < 1.96) =
0.5 - P(Z > 1.96) = 0.5 - 0.0250 = 0.4750
You can combine the above tips to find probabilities such as
P(-1 < Z < 2) =
1 - P(Z > 2) - P(Z < -1) =
1 - P(Z > 2) - P(Z > 1) =
1 - 0.1587 - 0.0228 = 0.8185
To find a probability like P(1 < Z < 2) you need to subtract
two probabilities. The table gives you P(Z > 1) = 0.1587 directly,
but this probability is the entire area under the curve to the right of 1. You need to
subtract from that the area to the right of 2 to get the desired probability. So,
P(1 < Z < 2) =
P(Z > 1) - P(Z > 2) =
0.1587 - 0.0228 = 0.1359.
