Download Uniform Distribution Notes

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Chapter 5-1
Chapter 5: Continuous Random Variables -Uniform Distribution
Example U1:
A package delivery service divides their packages into weight classes. Suppose that packages in the 14
to 20 pound class are uniformly distributed, meaning that all weights within that class are equally likely
to occur. We're interested in studying package weights in the 14 to 20 pound size.
a. Define the random variable, X = ______________________________________________________
 Notation for a uniformly distributed random variable with parameters of a minimum value
of a and a maximum value of b: X~U(a, b) .
b. Write the distribution of X. X~ _____________________________
 Understanding The Graph:
 The graph of the uniform distribution is a rectangle.
 The horizontal axis represents values of the random variable, X.
 The vertical axis represents values of the probability density function, f(x).
 The rectangle begins at the minimum x-value, a and ends at the maximum x-value b.
The total area of the rectangle equals the total probability of the variable, X.
So,
1 = area of rectangle
1 = base • height
1 = (b - a) • f(x)
1/ (b-a) = f(x)
A uniformly distributed random variable X has
a constant probability density function:
f(x) = a constant value!
c. Neatly sketch a fully labeled graph of X. Use a ruler!
 Finding the Mean (average) and Standard Deviation:
d. Find the mean and standard deviation to 4 decimal places for X.
Use proper symbols for these parameters.
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Chapter 5-2
 Finding Probabilities:



Write the initial probability statement.
Show the formula and calculations to find the answer to 4 decimal places.
Draw a sketch of the corresponding graph with both axes labeled and scaled, appropriate
shading, and key probabilities and X values identified and labeled.
Example U1 continued:

For problems e through j, assume that one package is randomly selected.
e. Find the probability that the package weighs between 15 and 16.5 pounds, inclusive.
P(15 ≤ X ≤ 16.5 ) = area of shaded rectangle
P(15 ≤ X ≤ 16.5 ) = area of shaded rectangle
1
P(15 ≤ X ≤ 16.5 ) = area of shaded rectangle
area of distribution rectangle
P(15 ≤ X ≤ 16.5 ) =
(base of shaded rectangle)(height of shaded rectangle)
(base of distribution rectangle)(height of distribution rectangle)
P(15 ≤ X ≤ 16.5 ) =
(base of shaded rectangle)(height of shaded rectangle)
(base of distribution rectangle)(height of distribution rectangle)
P(15 ≤ X ≤ 16.5 ) = (base of shaded rectangle)
(base of distribution rectangle)
P(15 ≤ X ≤ 16.5 ) = (16.5 – 15)
(20 – 14)
P(15 ≤ X ≤ 16.5 ) = 1.5
6
P(15 ≤ X ≤ 16.5 ) =
0.25
Alternative Approach:
P(15 ≤ X ≤ 16.5 ) = area of shaded rectangle
P(15 ≤ X ≤ 16.5 ) = (base of shaded rectangle)(height of shaded rectangle)
P(15 ≤ X ≤ 16.5 ) = (16.5 – 14)(1/6) = 1.5(1/6) = 0.25
f. Find the probability that a package weighs less than 15 pounds.
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Chapter 5-3
g. Find the probability that a package weighs at least 18 pounds.
h. Find the probability that a package weighs exactly 17 pounds.
i. Find the probability a package weighs at least 18 pounds given it weighs at least 15 pounds.
In your sketch, we know that the package is not less than 15 pounds, so cross out the area of the
rectangle related to X values between 14 and 15. The original rectangle with X between 14 and 20 has
been reduced to a rectangle with X between 15 and 20.
P(X ≥ 18 │ X ≥ 15) = area of shaded rectangle
P(X ≥ 18 │ X ≥ 15) = area of shaded rectangle
1
P(X ≥ 18 │ X ≥ 15) = area of shaded rectangle
area of reduced rectangle
P(X ≥ 18 │ X ≥ 15) = (base of shaded rectangle)(height of shaded rectangle)
(base of reduced rectangle)(height of reduced rectangle)
P(X ≥ 18 │ X ≥ 15) = (base of shaded rectangle)(height of shaded rectangle)
(base of reduced rectangle)(height of reduced rectangle)
P(X ≥ 18 │ X ≥ 15) = (base of shaded rectangle)
(base of reduced rectangle)
P(X ≥ 18 │ X ≥ 15)= (20 – 18)
(20 – 15)
P(X ≥ 18 │ X ≥ 15) = 2
5
P(X ≥ 18 │ X ≥ 15) = 0.40
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Chapter 5-4
j. Find the probability a package weighs at most 16 pounds given it weighs over 14.5 pounds.
Inverse Problems
k. Percentile area is on the left.
Find the 40th percentile of package weights in this weight class. Interpret your result.
Area left of percentile = P(X ≤ percentile) = decimal format of related percent
Area left of percentile = P(X ≤ k) = 0.40
base of shaded rectangle
= 0.40
base of distribution rectangle
k – 14
20 - 14
= 0.40
k – 14
= 0.40 (20 – 14)
k
= 14 + 0.40 (20 – 14)
k = 16.4 pounds
Alternatively:
The percentile, k, is a value on the X-axis. 40% of the area is to the left of k.
k, is the x-value which is 40% of the distance from the left edge of the outer rectangle to the right edge.
k = the left edge of the rectangle + 40% of the distance to the right edge of the outer rectangle.
k =
14
+ 0.40 (20 – 14)
k=
14
+
2.4
k = 16.4 pounds
P(X ≤ 16.4) = 0.40
Interpretation:
 The 40th percentile is 16.4 pounds.
OR
40 % of packages weigh at most 16.4 pounds.
l. Finding X when given area on the right:
20 % of packages weigh at least how much? Interpret your result.
(Find the minimum weight for the heaviest 20% of packages.)
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Chapter 5-5
Example U2:
According to the US Department of energy, commercial food freezers usually maintain a very low
temperature range. Suppose that temperature in the food freezers varies between – 4 F and 2 F,
following a uniform distribution. Answer probabilities to 3 decimal places. Answer temperatures to
tenths of a degree.
a. Define the random variable, X = ______________________________________________________
b. Show the notation for the distribution of X.
c. Draw a graph of the probability distribution with both axes labeled and scaled.
d. Find the mean of this distribution.
e. Find the median of this distribution.
f. Find the standard deviation of this distribution. Show your work.
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Chapter 5-6
For parts g through l, write the initial probability statement, show all calculations to reach a final
answer to 3 decimal places, and include a well labeled sketch of the situation.
a. Find the probability that the temperature is less than −2 F.
b. Find the probability that the between −1 F and 0.5 F.
c. Find the first quartile (Q1) for this distribution.
d. Find the temperature value, x, that makes this statement true:
“The temperature in the freezer is above x  F, 40% of the time.”
What percentile does this temperature represent? Interpret the result.
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e.
Chapter 5-7
Find the probability that the temperature of the freezer is no more than – 1.5  F, given the
temperature is less than 0  F.
f. Find the probability that the temperature of the freezer is at least 0  F, given that the temperature is
more than 1  F.
Exponential Distribution
The exponential distribution:
 is a continuous function
 is a decreasing function
 with a domain of X ≥ 0
The probability density function, f(x), is:
f ( x)  me mx , where m  0 and x  0 .
e = 2.71828182846, an irrational number.

If X is distributed exponentially, then the notation is: X  Exp (m).

m = the Parameter of Decay or Decay Rate

If X is distributed exponentially, then the notation is: X  Exp (m).

m and  are reciprocals:  

σ =  =

The area under the entire curve is 1.

To find the vertical intercept of the exponential function: f (0)  me  m  0  me 0  m  1  m .

The exponential distribution is skewed ________________.

Circle the correct relationship?
1
,
m
and
m
1

.
1
m
μ < M
,
μ = M
,
or
μ > M
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Chapter 5-8
The 4 formulas for finding areas under the exponential curve and for finding percentiles are:

Area to the left of x:
P( X  x)  1  e mx
 Area to the right of x:
P( X  x)  e mx
 Area between two values of x, a and b:
P(a  X  b )  e ma  e mb
 Inverse Formula: To find k, the nth percentile
k = ln (1 – the area to the left of k) ,
(-m)
where ln is the natural logarithm function.
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Chapter 5-9
Exponential Distribution Problems
Example E1:
Suppose that the length of long distance phone calls, in minutes, is known to have an exponential
distribution with the average length of a call equal to 8 minutes.
a. Define the random variable, X = ______________________________________________________
b. Write the symbol for the mean of the distribution. Calculate its value.
c. Write the symbol for the standard deviation of the distribution. Calculate its value.
d. Write the symbol for the decay parameter of the distribution. Calculate its value.
e. Write the notation for the distribution of the random variable, X.
f. Write the probability density function, using proper notation and symbols.
g. Sketch a fully labeled graph.
For parts h through m,
 Translate the problem with a probability statement.
 Write the formula for your calculations.
 Answer the problem with a decimal to 4 places.
 Draw the corresponding sketch, neatly, fully labeled, and with proper shading of areas.
h. Find the probability that a phone call lasts less than 9 minutes.
Last Name____________________ First Name _________________Class Time________ Chapter 5-10
i. Find the probability that a phone call lasts between 7 and 9 minutes.
j. Find the probability that a phone call lasts more than 5 minutes.
l. Find the median length of calls made. Include a sketch of the situation. Interpret your result in
context of the problem.
m. Find the 42nd percentile of the length of calls made. Include a sketch of the situation.
Interpret your result in context of the problem.
k. If 25 phone calls are made, one after another, how long would you expect the 25 calls to last?
o. Enter 66/100,000 into your calculator; write the result exactly as shown in your calculator window:
Now, estimate this result to 4 decimal places.