Normal Distribution
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N(μ, σ2)
Bell-shaped, unimodal, symmetric
Mean, μ, E(X), always in the center of the curve (the peak)
Variance, σ2, controls the spread of the graph (wide or narrow) – remember the standard deviation (σ)
is simply the square root of variance.
Probabilities are just the area under the curve (integral) between the points of interest
Total area under the Normal curve = 1 (or 100%)
Curve stretches from -∞ to + ∞ , but the area under the curve gets very small the farther you go from
the mean.
PDF = 𝑓(𝑥) =
1
√2𝜋𝜎
𝑒
−(
𝑥−𝜇 2
)
𝜎
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− ∞ < 𝑥 < ∞ **note we will not actually need to USE this formula**
Approximately 68% of the data will fall within 1σ of the mean
(between µ-1σ and µ+1σ).
Approximately 95% of the data will fall within 2σ of the mean
(between µ-2σ and µ+2σ).
Approximately 99.7% of the data will fall within 3σ of the mean
(between µ-3σ and µ+3σ).
The normal distribution does not have a general form of a CDF that we can define for any normal distribution
and just plug in our parameters – we only have a CDF for what we call the Standard Normal distribution:
𝑍~𝑁𝑜𝑟𝑚𝑎𝑙(𝜇 = 0, 𝜎 2 = 1)
So what we do, is similar to geometry and similar triangles. All Normal distributions will be “similar” (all bell
shaped) and so we can scale all other normal distributions in terms of z-scores (number of standard deviations
away from the mean).
Example: Exam scores for the last exam followed a Normal Distribution with a mean of 80 and variance of 25
(standard deviation of 5). Let X be your exam score.
P(X<80) = P(Z<0) **80 is 0 standard deviations away from the mean**
P(X>90) = P(Z>2) **90 is 2 standard deviations away from the mean**
We can do this with ANY normal distribution with any mean and any variance.
Given an “x” value (a test score in this example) – to find the number of standard deviations away from the
mean (z-score)
𝑧=
𝑥−𝜇
𝜎
Now that we can handle any normal distribution and find z-scores, how do we now find probabilities???
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4.
𝑥−𝜇
Convert X value to a z-score using the formula 𝑧 = 𝜎
Rearrange (if necessary) so inequality is < or ≤. Remember that P(Z>z-score) = 1 – P(Z ≤ z-score)
Look up the probability for your z-score in table.
If z-score is between 2 table values, either pick the closer one or average the closest values.
Practice:
a. 𝑃(𝑍 ≤ 1.24) =
b. 𝑃(𝑍 > 1.24) =
c. 𝑃(𝑍 ≤ − 1.24) =
d. 𝑃(𝑍 > − 1.24) =
e. 𝑃(𝑍 = 1.24) =
f. 𝑃(− 1.24 ≤ 𝑍 ≤ 1.24) =
g. 𝑃(𝑍 ≤ −4.5) =
h. 𝑃(𝑍 ≤ 12) =
Example: Now practice using the Normal table with some generic problems where you start with X and then
have to find Z before going to the table. For all the probabilities: 𝑋~𝑁𝑜𝑟𝑚𝑎𝑙(𝜇 = 2, 𝜎 2 = 1.44)
a. 𝑃(𝑋 < 1) =
b. 𝑃(𝑋 > 4) =
c. 𝑃(0 < 𝑋 < 3.4) =
d. 𝑃(𝑋 = 2.4) =
e. 𝑃(𝑋 < 12) =
Example: Nancy works at PEFCU and tells us that checking account balances follow ~𝑁(µ = 1325, 𝜎 2 = 625)
. Bill has a balance of $1270.
a. What is the probability an account will have less money than Bill’s?
b. What is the probability an account balance will be more than $1280?
c. What is the probability an account balance will be exactly $1380?
d. What is the probability an account will have less than $1325 (the mean)?
e. What is the probability that an account will have between $1310 and $1390?
f. What is the probability an account will have less than $10?
“Backwards” Normal Distribution
If you are given the probability and know X~N(µ,σ2), but don’t know the x value that would give that
probability.
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Convert to P(Z<z-score) = probability (you may need to do 1 – if you’re looking for > prob).
Find the probability that closest meets the probability you’re looking for.
Find the corresponding z-score that would give the specified probability.
Convert the z-score to x using x = µ + zσ
Practice: Use the normal table with some generic backwards-normal problems where we know the probability
and are just looking for z0 (instead of going all the way to x). Find the following z-scores:
a. 𝑃(𝑍 ≤ 𝑧0 ) = 0.5000
b. 𝑃(𝑍 ≤ 𝑧0 ) = 0.9953
c. 𝑃(𝑍 ≤ 𝑧0 ) = 0.2206
d. 𝑃(𝑍 ≤ 𝑧0 ) = 0.95
e. 𝑃(𝑍 > 𝑧0 ) = 0.0129
f. 𝑃(𝑍 > 𝑧0 ) = 0.5025
g. 𝑃(𝑧0 ≤ 𝑍 ≤ 𝑧0 ) = 0.20
Example: Practice using normal table with some generic backwards-normal problems where we know the
probability and are now going all the way to x0 by way of z0. For all problems: 𝑋~𝑁𝑜𝑟𝑚𝑎𝑙(𝜇 = 2, 𝜎 2 = 1.44)
a. 𝑃(𝑋 ≤ 𝑥0 ) = 0.2206
b. 𝑃(𝑋 > 𝑥0 ) = 0.95
c. Find the 2 values between which 20% of the data lies.
Example: In the checking account example where the balances are ~𝑁(µ = 1325, 𝜎 2 = 625)
a. What is the account balance, x0, such that the percentage of balances less than it is 23%?
b. What is the account balance, x0, such that the probability of a balance being more than it is 0.15?
c. What is the account balance, x0, such that the probability of a balance being more than it is 0.5? (Hint:
you should be able to do this one without math)
d. Between what 2 central values do 40% of the balances fall?