Respuesta :
Answer:
(a) The mean is 4.31 pounds. The variance is 51.42 pounds.
(b) The average shipping cost of a package is $10.78.
(c) The probability that the weight of a package exceeds 59 pounds is 0.0027.
Step-by-step explanation:
The probability density function of the weight of packages is:
[tex]f(x) = \frac{70}{69x^{2}};\ 1 < x < 70[/tex]
(a)
The formula for expected value (or mean) of X is:
[tex]E(X)=\int\limits^a_b {x\times f(x)} \, dx[/tex]
Compute the expected value of X as follows:
[tex]E(X)=\int\limits^{70}_{1} {x\times \frac{70}{69x^{2}}} \, dx=\frac{70}{69} \int\limits^{70}_{1} {x \times x^{-2}} \, dx\\=\frac{70}{69} \int\limits^{70}_{1} {x^{-1}} \, dx=\frac{70}{69} |\ln x|^{70}_{1}\\=\frac{70}{69}\times\ln 70\\=4.31[/tex]
Thus, the mean is 4.31 pounds.
The formula to compute the variance is:
[tex]V(X)=E(X^{2})-[E(X)]^{2}[/tex]
Compute the E (X²) as follows:
[tex]E(X^{2})=\int\limits^{70}_{1} {x^{2}\times \frac{70}{69x^{2}}} \, dx=\frac{70}{69} \int\limits^{70}_{1} {x^{2} \times x^{-2}} \, dx\\=\frac{70}{69} \int\limits^{70}_{1} {1} \, dx=\frac{70}{69} | x|^{70}_{1}\\=\frac{70}{69}\times69\\=70[/tex]
The variance is:
[tex]V(X)=E(X^{2})-[E(X)]^{2}\\=70-(4.31)^{2}\\=51.4239\\\approx51.42[/tex]
Thus, the variance is 51.42 pounds.
(b)
It is provided that the shipping cost for per pound is, C = $2.50.
Compute the average shipping cost of a package as follows:
[tex]Average\ cost=Cost\ per\ pound\times E(X)\\=2.50\times4.31\\=10.775\\\approx10.78[/tex]
Thus, the average shipping cost of a package is $10.78.
(c)
Compute the probability that the weight of a package exceeds 59 pounds as follows:
[tex]P(59<X<70)=\int\limits^{70}_{59} {\frac{70}{69x^{2}}} \, dx=\frac{70}{69} \int\limits^{70}_{59} {x^{-2}} \, dx\\=\frac{70}{69} |-\frac{1}{x}|^{70}_{59}=\frac{70}{69} [-\frac{1}{70}+\frac{1}{59}]\\=\frac{70}{69}\times0.0027\\=0.0027[/tex]
Thus, the probability that the weight of a package exceeds 59 pounds is 0.0027.
The weights of the package follows a probability density function
- The mean is 4.31 and the variance is 51.42, respectively.
- The average cost of shipping a package is $10.78
- The probability a package weighs over 59 pounds is 0.0027
The probability density function is given as:
[tex]\mathbf{f(x) = \frac{70}{69x^2},\ 1 < x < 70}[/tex]
(a) The mean and the variance
The mean is calculated as:
[tex]\mathbf{E(x) = \int\limits^a_b {x \cdot f(x)} \, dx }[/tex]
So, we have:
[tex]\mathbf{E(x) = \int\limits^{70}_1 {x \cdot \frac{70}{69x^2} } \, dx }[/tex]
[tex]\mathbf{E(x) = \int\limits^{70}_1 {\frac{70}{69x} } \, dx }[/tex]
Rewrite as:
[tex]\mathbf{E(x) = \frac{70}{69}\int\limits^{70}_1 {x^{-1} } \, dx }[/tex]
Integrate
[tex]\mathbf{E(x) = \frac{70}{69} {ln(x)}|\limits^{70}_1 } }[/tex]
Expand
[tex]\mathbf{E(x) = \frac{70}{69} \cdot {(ln(70) - ln(1)) }}[/tex]
[tex]\mathbf{E(x) = 4.31 }}[/tex]
The variance is calculated as:
[tex]\mathbf{Var(x) = E(x^2) - (E(x))^2}[/tex]
Where:
[tex]\mathbf{E(x^2) = \int\limits^a_b {x^2 \cdot f(x)} \, dx }[/tex]
So, we have:
[tex]\mathbf{E(x^2) = \int\limits^{70}_1 {x^2 \cdot \frac{70}{69x^2} } \, dx }[/tex]
[tex]\mathbf{E(x^2) = \int\limits^{70}_1 {\frac{70}{69} } \, dx }[/tex]
Rewrite as:
[tex]\mathbf{E(x^2) = \frac{70}{69}\int\limits^{70}_1 {1 } \, dx }[/tex]
Integrate
[tex]\mathbf{E(x^2) = \frac{70}{69} x|\limits^{70}_1 } }[/tex]
Expand
[tex]\mathbf{E(x^2) = \frac{70}{69} \cdot {(70 - 1) }}[/tex]
[tex]\mathbf{E(x^2) = 70 }}[/tex]
So, we have:
[tex]\mathbf{Var(x) = E(x^2) - (E(x))^2}[/tex]
[tex]\mathbf{Var(x) = 70 - 4.31^2}[/tex]
[tex]\mathbf{Var(x) = 51.42}[/tex]
Hence, the mean is 4.31 and the variance is 51.42, respectively.
(b) The average cost of shipping a package
In (a), we have:
[tex]\mathbf{E(x) = 4.31 }}[/tex] ---- Mean
So, the average cost of shipping a package is:
[tex]\mathbf{Average =4.31 \times 2.50}[/tex]
[tex]\mathbf{Average =10.78}[/tex]
Hence, the average cost of shipping a package is $10.78
(c) The probability a package weighs over 59 pounds
This is represented as: P(x > 59)
So, we have:
[tex]\mathbf{P(x > 59) = P(59 < x < 70)}[/tex]
So, we have:
[tex]\mathbf{P(x > 59) = \int\limits^{70}_{59} { \frac{70}{69x^2}} \, dx }[/tex]
Rewrite as:
[tex]\mathbf{P(x > 59) = \frac{70}{69}\int\limits^{70}_{59} { x^{-2}} \, dx }[/tex]
Integrate
[tex]\mathbf{P(x > 59) = \frac{70}{69} \cdot { -\frac 1x}|\limits^{70}_{59}}[/tex]
Expand
[tex]\mathbf{P(x > 59) = \frac{70}{69} \cdot (-\frac{1}{70} + \frac{1}{59})}[/tex]
[tex]\mathbf{P(x > 59) = 0.0027}[/tex]
Hence, the probability a package weighs over 59 pounds is 0.0027
Read more about probability density functions at:
https://brainly.com/question/14749588
