HW5_urdaneta
pdf
School
Virginia Tech *
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Course
2024
Subject
Industrial Engineering
Date
May 2, 2024
Type
Pages
5
Uploaded by CommodoreMandrill4162
ISE 2024 - Homework 5
Due on April 10th, 11:59pm
This assignment contains 6 questions for a total of 100 points.
Note that submission is on Gradescope
.
1.
(16 points) If Z is N(0, 1), find
•
P
(0 ≤ Z ≤ 0
.
87) = 0.3133
•
P
(−2
.
64 ≤ Z ≤ 0) P
(−2
.
64 ≤ Z ≤ 0) = 0.4953
•
P
(−2
.
13 ≤ Z ≤ −0
.
56) = 0.2702
•
P
(|
Z
| > 1
.
39)
. = 0.8354
2.
(20 points) Consider the random variable X as the location of the next online delivery order requested by a customer. If X is normally distributed with a mean of 6 and a variance of 25,
Find
Standard Deviation = sqrt(variance) = 5
•
P
(6 ≤ X ≤ 12) = 0.3849
Standardizing: z₁ = (6 - 6) / 5 = 0 z₂ = (12 - 6) / 5 = 1.2
P(0 ≤ Z ≤ 1.2)
•
P
(0 ≤ X ≤ 8) = 0.4918
Standardizing: z₁ = (0 - 6) / 5 = -1.2 z₂ = (8 - 6) / 5 = 0.4
P(-1.2 ≤ Z ≤ 0.4)
•
P
(−2 < X ≤ 0)
. = 0.1534
Standardizing: z₁ = (-2 - 6) / 5 = -1.6 z₂ = (0 - 6) / 5 = -1.2
P(-1.6 < Z ≤ -1.2)
•
P
(|
X − 6| < 5)
. = 0.6827
P(1 < X < 11)
Standardizing: z₁ = (1 - 6) / 5 = -1 z₂ = (11 - 6) / 5 = 1
P(-1 < Z < 1)
3.
(14 points) The strength X of a certain material is such that its distribution is found by X = e
Y , where Y is N
(10
,
1). Find the CDF and PDF of X
, and compute P
(10
,
000 < X < 20
,
000). Note: F
(
x
) = P
(
X ≤ x
) = P
(
e
Y ≤ x
) = P
(
Y ≤ ln
x
) so that the random variable X is said to have a lognormal distribution
.
F(x)=P(X≤x)=P(e
Y
≤x)=P(Y≤lnx)
P(10000≤X≤20000)=P(ln(10000)≤Y≤ln(20000))≈P(−0.7897≤(Y−10)≤−0.0965)
≈0.24692
4.
(20 points) Assume that X ∼ N(
µ,σ
2
). Show that Var(
X
) = σ
2
.
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5.
(10 points) The lifetime in hours of an electronic tube is a random variable having a probability density function given by
f
(
x
) = xe
−
x
,x ≥ 0
.
Compute the expected lifetime of such a tube.
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6.
(20 points) In this question we aim at generating Gaussian Random variables. For this, we use Box-Muller transform method. Use Excel (or any other software that you feel comfortable using) and generate 1000 uniform(0
,
1) RVs, 500 of which is denoted as U
1 and 500 others denoted as U
2
. Then use the following formula to generate 1000 samples of standard normal
variables:
.
Verify your simulation with plotting the histogram of the generated gaussian random variables.
Include all your data points and the plots in your solution.
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