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Chapter 5. Continuous Probability
Distributions
Sections 5.2, 5.3: Expected Value of Continuous
Random Variables and Uniform Distribution
Jiaping Wang
Department of Mathematical Science
03/20/2013, Monday
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Outline
EV: Definitions and Theorem
EV: Examples
Uniform Distribution: Density and Distribution
Functions
Uniform Distribution: Mean and Variance
More Examples
Homework #8
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Part 1. EV: Definitions and
Theorem
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Definition and Theorem
Definition 5.3: The expected value of a continuous random
variable X that has density function f(x) is given by
∞
𝐸 𝑋 =
𝑥𝑓 𝑥 𝑑𝑥 .
−∞
Note: we assume the absolute convergence of all integrals so
that the expectations exist.
Theorem 5.1: If X is a continuous random variable with
probability density f(x), and if g(X) is any real-valued function of
∞
X, then
𝐸 𝑔 𝑋 = −∞ 𝑔 𝑥 𝑓 𝑥 𝑑𝑥
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Variance
Definition 5.4: For a random variable X with probability density
function f(x), the variance of X is given by
∞
V 𝑋 = 𝐸 𝑋 − 𝜇 2 = −∞ 𝑥 − 𝜇 2𝑓 𝑥 𝑑𝑥 = 𝐸 𝑋2 − 𝜇2 .
Where μ=E(X).
For constants a and b, we have
E(aX+b)=aE(X)+b
V(aX+b)=a2V(X)
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Part 2. EV: Examples
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Example 5.4
For a given teller in a bank, let X denote the proportion of time,
out of a 40-hour workweek, that he is directly serving customers.
Suppose that X has a probability density function given by
3𝑥2, 0 ≤ 𝑥 ≤ 1
f x =
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
1. Find the mean proportion of time during a 40-hour workweek
the teller directly serve customers.
2. Find the variance of the proportion of time during a 40-hour
workweek the teller directly serves customers.
3. Find an interval that, for 75% of the weeks, contains the
proportion of time that the teller spends directly serving
customers.
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Answer: 1. Based on the definition,
∞
𝐸 𝑋 =
1
𝑥 3𝑥2 𝑑𝑥 = 3/4.
𝑥𝑓 𝑥 𝑑𝑥 =
−∞
0
Thus, on average, the teller spends 75% of his time each week directly serving
customers.
2. We need to compute the E(X2):
∞
1
𝑥2𝑓 𝑥 𝑑𝑥 =
𝐸 𝑋 =
−∞
𝑥2 3𝑥2 𝑑𝑥 = 0.60.
0
V(X)=E(X2)-E2(X)=0.60-(0.75)2=0.0375.
Then,
3. There are lots of ways to construct the interval such that the proportion of time
that the teller spends directly serving customers for 75% of the weeks, for
example, P(X<a)=0.12, P(X>b)=0.13, or P(X<a)=0.10, P(X>b)=0.15, for the other
25% of the weeks. We choose the half of 25% for the two sided tails, ie.,
P(X<a)=0.125 and P(X>b)=0.125 for some a and b. So we have
P(X<a)=a3=0.125a=0.5, P(X>b)=1-b3=0.125b=0.956. That is, for 75% of the
weeks, the teller spends between 50% and 95.6% of his time directly serving
customers.
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Example 5.5
The weekly demand X, in hundreds of gallons, for propane at a certain supply
station has a density function given by
𝑥
,0 ≤ 𝑥 ≤ 2
4
𝑓 𝑥 = 1
,2 < 𝑥 ≤ 3
2
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
It takes $50 per week to maintain the supply station. Propane is purchased for
$270 per hundred gallons and redistributed by the supply station for $1.75 per
gallon.
1. Find the expected weekly demand.
2. Find the expected weekly profit.
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Answer: 1. Based on the definition,
∞
𝐸 𝑋 =
2
𝑥𝑓 𝑥 𝑑𝑥 =
−∞
𝑥
0
4
𝑑𝑥 +
𝑥
3
𝑥
2
1
𝑑𝑥 = 1.92.
2
Thus, on average, the weekly demand for propane will be 192 gallons at this
supply station.
2. The propane is purchased for $270 per hundred gallons and sold for $175 per
hundred gallons, yielding a profit of $95 per hundred gallons sold. The weekly
profit P is given as P=95X-50, so E(P)=95E(X)-50=95(1.92)-50=132.40.
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Tchebysheff’s Theorem and Example 5.6
The Tchebysheff’s theorem holds for the continuous random variable, X, ie.,
P(|X-μ|<kσ) ≥ 1-1/k2
Example 5.6: The weekly amount X spent for chemicals by a certain firm has a
mean of $1565 and a variance of $428. Within what interval should these
weekly costs for chemicals be expected to lie in at least 75% of the time?
Answer: To find the interval guaranteed to contain at least 75% of the
probability mass for X, we need to have 1-1/k2=0.75  k=2.
So the interval is given by [1565-2(428)1/2, 1565+2(428)1/2].
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Part 3. Uniform Distribution:
Density and Distribution
Functions
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Density Function
Consider a simple model for the continuous random variable X, which is
equally likely to lie in an interval, say [a, b], this leads to the uniform
probability distribution, the density function is given as
𝟏
𝒇 𝒙 = 𝒃 − 𝒂,𝟎 ≤ 𝒙 ≤ 𝒃
𝟎,
𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
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Cumulative Distribution Function
The distribution function for a uniformly distributed X is given by
0, 𝑥 < 1
𝑥−𝑎
, 𝑎≤𝑥≤𝑏
𝐹 𝑥 =
𝑏−𝑎
1, 𝑥 > 𝑏
For (c, c+d) contained within (a, b), we have
P(c≤X≤c+d)=P(X≤c+d)-P(X≤c)=F(c+d)-F(c)=d/(b-a), which this
probability only depends on the length d.
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Mean and Variance
∞
𝐸 𝑋 =
𝑏
𝑥𝑓 𝑥 𝑑𝑥 =
−∞
𝑎
∞
𝐸 𝑋2 =
𝑏
𝑥2𝑓 𝑥 𝑑𝑥 =
−∞
𝑉 𝑋 = 𝐸 𝑋2 − 𝐸2 𝑋 =
𝑎
𝑎2+𝑎𝑏+𝑏2 𝑎+𝑏
3
2
the length of the interval [a, b].
𝑥
𝑎+𝑏
𝑑𝑥 =
𝑏−𝑎
2
𝑥2
𝑎2 + 𝑎𝑏 + 𝑏2
𝑑𝑥 =
𝑏−𝑎
3
2=
1
12
𝑏 − 𝑎 2 which depends only on
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Example 5.7
A farmer living in western Nebraska has an irrigation system to provide water
for crops, primarily corn, on a large farm. Although he has thought about
buying a backup pump, he has not done so. If the pump fails, delivery time X
for a new pump to arrive is uniformly distributed over the interval from 1 to 4
days. The pump fails. It is a critical time in the growing season in that the yield
will be greatly reduced if the crop is not watered within the next 3 days.
Assuming that the pump is ordered immediately and the installation time is
negligible, what is the probability that the farmer will suffer major yield loss?
Answer: Let T be the time until the pump is delivered. T is uniformly distributed over
The interval [1, 4]. The probability of major loss is the probability that the time until
Delivery exceeds 3 days. So
4
1
1
𝑃 𝑇>3 =
𝑑𝑥 = .
3
3 3
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Part 3. More Examples
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Additional Example 1
Let X have the density function given by
1.
2.
3.
4.
Find the value c.
Find F(x).
P(0≤X≤0.5).
E(X).
Answer:
∞
1. −∞ 𝑓 𝑥 𝑑𝑥 = 1 →
0
0.2𝑑𝑥
−1
+
1
0
𝑐
0.2 + 𝑐𝑥 𝑑𝑥 = 1 → 0.2 + 0.2 + 2 = 1 → 𝑐 = 1.2
0, 𝑥 ≤ −1
0.2𝑥 + 0.2, −1 ≤ 𝑥 ≤ 0
2. 𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥 =
3. P(0≤X≤0.5)=0.25,
0.2 + 0.2𝑥 + 0.6𝑥2, 0 < 𝑥 ≤ 1
1, 𝑥 > 1
4. E(X)=0.4
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Additional Example 2
Let X have the density function
Find E(lnX).
Answer: 𝐸 𝑙𝑛𝑋 =
∞
ln
−∞
𝑥 𝑓 𝑥 𝑑𝑥 =
𝑒 ln 𝑥
1 𝑥
1
𝑑𝑥 = 2 .
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Homework #8
Page 199-200: 5.4, 5.7
Page 209: 5.22
Page 214-215: 5.28, 5.40.
Additional Hw1: Let X have the density function
Find the E(X).
Additional Hw2: The density function of X is given by
(a). Find a and b.
(b). Determine the cumulative distribution function F(x).
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