Challenge Problem In statistics, the standard normal density function is given by f ( x ) = 1 2 π ⋅ exp [ − x 2 2 ] . This function can be transformed to describe any general normal distribution with mean μ , and standard deviation, σ . A general normal density function is given by f ( x ) = 1 2 π ⋅ σ ⋅ exp [ − ( x − μ ) 2 2 σ 2 ] . Describe the transformations needed to get from the graph of the standard normal function to the graph of a general normal function.
Challenge Problem In statistics, the standard normal density function is given by f ( x ) = 1 2 π ⋅ exp [ − x 2 2 ] . This function can be transformed to describe any general normal distribution with mean μ , and standard deviation, σ . A general normal density function is given by f ( x ) = 1 2 π ⋅ σ ⋅ exp [ − ( x − μ ) 2 2 σ 2 ] . Describe the transformations needed to get from the graph of the standard normal function to the graph of a general normal function.
Solution Summary: The author explains the transformations needed to graph the general normal function f(x)=1sqrt
Challenge Problem In statistics, the standard normal density function is given by
f
(
x
)
=
1
2
π
⋅
exp
[
−
x
2
2
]
. This function can be transformed to describe any general normal distribution with mean
μ
,
and standard deviation,
σ
. A general normal density function is given by
f
(
x
)
=
1
2
π
⋅
σ
⋅
exp
[
−
(
x
−
μ
)
2
2
σ
2
]
. Describe the transformations needed to get from the graph of the standard normal function to the graph of a general normal function.
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.
(USE R-Studio to solve this question)
Concentration of pollutants produced by chemical plants historically are known to exhibit behavior that resembles a log normal distribution. This is important when one considers issues regarding compliance to government regulations. Suppose it is assumed that the concentration of a certain pollutant, in parts per million, has a lognormal distribution with parameters θ=3.6 and ω = 0.5
a)Is this a valid probability density function? What is the relationship between the lognormal r.v. and the corresponding normal r.v.? Explain briefly.
b)What is the probability that the concentration exceeds 15 parts per million? Illustrate the desired probability on the lognormal pdf plot. Calculate the same probability by using normal distribution and illustrate the probability on the normal pdf plot.
c)What is the probability that the concentration is between 8 and 16 parts per million? Illustrate the desired probability on the lognormal pdf plot. Calculate the…
Martian potatoes begin to sprout very quickly after planting. Suppose X is the number of days after planting until a Martian potato sprouts. Then X has the following probability density function: f(x)=
2/7e−x
+
3/14e−x/2
+
1/14e−x/4
for
0 ≤ x ≤ ∞
and 0 otherwise.
f) What is the variance of X? g) What is the standard deviation of X? h) What is the probability that X is more than 2 standard deviations above its expected value?
6) There are many engineering properties which cannot be negative, and so cannot be modeled using a normal distribution (since the normal distribution always has some probability of being negative). One such engineering property is the tensile strength of glue – it doesn’t make sense that a glue’s tensile strength would be negative. A common non-negative distribution used in engineering models is the lognormal, which is strictly non-negative and has a simple relationship with the normal distribution. Suppose that a particular type of glue is is lognormally distributed with mean 10 MPa and coefficient of variation 25%. If a random sample of this glue is tested,
a) What is the probability that its tensile strength will lie between 6 and 12 MPa?
thglue’s design tensile strength? (Note: the 10th percentile is the value, x10, such that
b) Suppose that the design tensile strength of this glue is its 10
percentile. What is this
??
PX<x10 =0.1)
Calculus Early Transcendentals, Binder Ready Version
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.