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Statistical Distributions and Moment Generating Functions - Prof. Lawrence Herman Winner, Exams of Probability and Statistics
A series of statistical problems related to various distributions, such as gamma, normal, uniform, exponential, beta, and gamma distributions. It covers topics such as moment generating functions, means, medians, and expected monetary values. The problems require the use of statistical formulas and properties to find solutions.
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STA 4321/5325 – Exam 3 – Spring 2007
( )
( ) ( )
( ) ( 1 )
1
0
1 1
0
1 /
y e dy y y dy
y
The moment generating function for the gamma distribution with parameters and is
M(t) = (1-t)
-
. Use this to demonstrate that the mean is in fact .
M’(t) = - (1- t)
-
) E(Y) = M’(0) =
People are dispersed on a linear beach with a density function f(y)=4y
3
0 < y < 1, 0
elsewhere. An ice cream vendor wishes to locate her cart at the median of the locations
(where half of the people will be on each side of her). Where will she locate her cart?
- 5 4 0. 5. 8409
4 1 / 4
0
4
0
3
y dy y m m
m m
Scores on a standardized exam are normally distributed with mean 1000 and standard
deviation 160.
What proportion of students score under 850 on the exam?
P(Y<850) = P(Z<(850-1000)/160 = -0.94) = .5-.3264 =.
They wish to calibrate the exam so that 1400 represents the 98
th
percentile. What
should they set the mean to (without changing the standard deviation).
P(Z<2.055) .98 P(Y < 1000+2.055(160) = 1328.8) .
Derive E(Y
k
) for the Uniform(0,1) distribution.
( 1 )
1
0
1
0
1
k k
y
EY y dy
k k k
The amount of time between arrivals at a car wash is exponentially distributed with a
mean of 30 minutes. What is the probability over 15 minutes passes between arrivals?
0 ( ). 6065
30
1 ( 15 )
1 / 2
15
/ 30
15
/ 30
P Y e dy e e
y y
Derive the moment generating function for the exponential distribution with mean .
0 1
0
0
/
0
( 1 )
exp 1
exp
( )
^
^
^
e e t t
y t
t
M t e e dy y t dy
ty y
A computer program generates pseudo-random numbers from a Uniform(0,1)
distribution. If it is working properly, what is the probability it will return 4 consecutive
numbers greater than 0.75?
(.25)
4 =.
The amount of impurity in batches of a chemical follow a Beta distribution with =1, and
=2.
Give the density function, as well as the mean amount of impurity.
f(y) = 2(1-y) 0≤ y ≤ 1
E(Y) = 1/(1+2) = 1/
If the amount of impurity exceeds 0.7, the batch cannot be sold. What proportion
of batches can be sold?
2 ( 1 ) 2 1. 4 0. 49 0. 91
- 7
0
2
- 7
0
ydy y y
Lifetimes of electrical components follow a Gamma distribution with =3, and =
Give the density function(be very specific to any numbers), as well as the mean
and standard deviation of the lifetimes.
f(y) = (1/432)y
2 e
-y/ y>
E(Y) = 18
Y = (3(6)
2 ) = 10.
The monetary value to the firm of a component with a lifetime of Y is V = 3Y
2
Y – 1. Give the expected monetary value of a component.
E(Y
2 ) = 108+(18)
2 = 432 E(V) = 3(432) + 18 -1 = 1313