Let f ( x ) denote the probability density function of a normal random variable with mean μ , and variance σ 2 . Show that μ − σ and μ + σ are points of inflection of this function. That is, show that f ' ' ( x ) = 0 when x = μ − σ or x = μ + σ .
Let f ( x ) denote the probability density function of a normal random variable with mean μ , and variance σ 2 . Show that μ − σ and μ + σ are points of inflection of this function. That is, show that f ' ' ( x ) = 0 when x = μ − σ or x = μ + σ .
Solution Summary: The author explains the probability density function f(x) of a normal random variable with mean and variance.
Let
f
(
x
)
denote the probability density function of a normal random variable with mean
μ
, and variance
σ
2
. Show that
μ
−
σ
and
μ
+
σ
are points of inflection of this function. That is, show that
f
'
'
(
x
)
=
0
when
x
=
μ
−
σ
or
x
=
μ
+
σ
.
Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
Answer questions 8.3.3 and 8.3.4 respectively
8.3.4 .WP An article in Medicine and Science in Sports and
Exercise [“Electrostimulation Training Effects on the Physical Performance of Ice Hockey Players” (2005, Vol. 37, pp.
455–460)] considered the use of electromyostimulation (EMS) as
a method to train healthy skeletal muscle. EMS sessions consisted of 30 contractions (4-second duration, 85 Hz) and were carried
out three times per week for 3 weeks on 17 ice hockey players.
The 10-meter skating performance test showed a standard deviation of 0.09 seconds. Construct a 95% confidence interval of the
standard deviation of the skating performance test.
8.6.7 Consider the tire-testing data in Exercise 8.2.3. Compute a 95% tolerance interval on the life of the tires that has confidence level 95%. Compare the length of the tolerance interval with the length of the 95% CI on the population mean. Which interval is shorter? Discuss the difference in interpretation of these two intervals.
8.6.2 Consider the natural frequency of beams described in
Exercise 8.2.8. Compute a 90% prediction interval on the
diameter of the natural frequency of the next beam of this type
that will be tested. Compare the length of the prediction interval
with the length of the 90% CI on the population mean.
8.6.3 Consider the television tube brightness test described in
Exercise 8.2.7. Compute a 99% prediction interval on the brightness of the next tube tested. Compare the length of the prediction
interval with the length of the 99% CI on the population mean.
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