Download probability density function

7.1
continuous random variables
(probability will be area under a curve)
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Uniform Distribution
EX: You are expecting a phone call during the next hour. The call could come at
any time (x) during the hour, with any time interval being equally likely. Thus, 0 ≤
x ≤ 60. Note: this is a uniform probability distribution (intervals of the same length
are equally likely).
Graph this distribution. Note the curve, y = 1/60, is the probability density function
(pdf).
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Probability density function (pdf)
1. Area under the curve = 1.
2. f(x) ≥ 0 for all x.
The area under the graph of a pdf over an interval represents the probability of
observing a value of the random variable in that interval.
Normal Distribution
(bell-shaped)
Properties of a Normal Distribution
1.
2.
3.
4.
5.
Symmetric about the mean. (mean = median)
Inflection points at μ-σ and μ+σ.
Area under the curve is 1. (Areas to the left and right of μ are each ½)
Horizontal asymptotes – horizontal axis.
Empirical Rule (68%, 95%, 99.7%).
Graph:
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Create/compare graphs:
a) μ = 0, σ = 3 (note actual graphs would have different heights, spread, etc.)
b) μ = 5, σ = 0.2
c) μ = 100, σ = 15
(interpret a score of 115, proportion or probability)
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Example: Assume IQ scores have a normal distribution with μ = 100 and σ = 15.
Use the Empirical Rule to find the following probabilities.
a) Find P(X > 100)
compare to P(X ≥ 100)
b) Find P(85 < X < 115)
c) Find P(X > 130)
d) Find P(X < 85)
Use the graph:
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Standardizing a Normal Random Variable
Suppose X is normally distributed with mean, μ, and standard deviation, σ.
Z=
𝑋−𝜇
𝜎
this variable is normally distributed with μ = 0 and σ = 1.
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Example: 1st: Find z-scores for IQ’s of 115, 130, 85, 106, 140.
2nd:
a) Find P(Z > 0)
b) Find P(-1 < Z < 1)
c) Find P(Z > 2)
d) Find P(Z < -1)
Repeat using normalcdf on calculator (note use of 1E99)
Also find: e) P(-0.4 < Z < 1.2)
f) P(Z < 0.46)
g) P(Z > -2.15)
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Note < vs. ≤ for continuous distributions.
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