Graphing them and see what the difference is
(i) The mean is
![\displaystyle E(X) = \sum_x x \, P(X = x) \\\\ E(X) = 1\cdot0.175 + 2\cdot0.315 + 3\cdot0.211 + 4\cdot0.092 + 5\cdot0.207 \\\\ \boxed{E(X) = 2.839}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20E%28X%29%20%3D%20%5Csum_x%20x%20%5C%2C%20P%28X%20%3D%20x%29%20%5C%5C%5C%5C%20E%28X%29%20%3D%201%5Ccdot0.175%20%2B%202%5Ccdot0.315%20%2B%203%5Ccdot0.211%20%2B%204%5Ccdot0.092%20%2B%205%5Ccdot0.207%20%5C%5C%5C%5C%20%5Cboxed%7BE%28X%29%20%3D%202.839%7D)
The variance is
![V(X) = E((X - E(X))^2) = E(X^2) - E(X)^2](https://tex.z-dn.net/?f=V%28X%29%20%3D%20E%28%28X%20-%20E%28X%29%29%5E2%29%20%3D%20E%28X%5E2%29%20-%20E%28X%29%5E2)
Compute the second moment
:
![\displaystyle E(X^2) = \sum_x x^2 \, P(X = x) \\\\ E(X) = 1^2\cdot0.175 + 2^2\cdot0.315 + 3^2\times0.211 + 4^2\times0.092 + 5^2\times0.207 \\\\ E(X^2) = 9.997](https://tex.z-dn.net/?f=%5Cdisplaystyle%20E%28X%5E2%29%20%3D%20%5Csum_x%20x%5E2%20%5C%2C%20P%28X%20%3D%20x%29%20%5C%5C%5C%5C%20E%28X%29%20%3D%201%5E2%5Ccdot0.175%20%2B%202%5E2%5Ccdot0.315%20%2B%203%5E2%5Ctimes0.211%20%2B%204%5E2%5Ctimes0.092%20%2B%205%5E2%5Ctimes0.207%20%5C%5C%5C%5C%20E%28X%5E2%29%20%3D%209.997)
Then the variance is
![\boxed{V(X) \approx 1.9171}](https://tex.z-dn.net/?f=%5Cboxed%7BV%28X%29%20%5Capprox%201.9171%7D)
(ii) For a random variable
, where
are constants, we have
![E(Z) = E(aX+b) = E(aX) + E(b) = a E(X) + b](https://tex.z-dn.net/?f=E%28Z%29%20%3D%20E%28aX%2Bb%29%20%3D%20E%28aX%29%20%2B%20E%28b%29%20%3D%20a%20E%28X%29%20%2B%20b)
and
![V(Z) = E((aX+b)^2) - E(aX+b)^2 \\\\ V(Z) = E(a^2 X^2 + 2ab X + b^2) - (a E(X) + b)^2 \\\\ V(Z) = a^2 (E(X^2) - E(X)^2) \\\\ V(Z) = a^2 V(X)](https://tex.z-dn.net/?f=V%28Z%29%20%3D%20E%28%28aX%2Bb%29%5E2%29%20-%20E%28aX%2Bb%29%5E2%20%5C%5C%5C%5C%20V%28Z%29%20%3D%20E%28a%5E2%20X%5E2%20%2B%202ab%20X%20%2B%20b%5E2%29%20-%20%28a%20E%28X%29%20%2B%20b%29%5E2%20%5C%5C%5C%5C%20V%28Z%29%20%3D%20a%5E2%20%28E%28X%5E2%29%20-%20E%28X%29%5E2%29%20%5C%5C%5C%5C%20V%28Z%29%20%3D%20a%5E2%20V%28X%29)
Then for
, we have
![E(Y) = \dfrac12 E(X) + \dfrac32 \\\\ \boxed{E(Y) = 2.918}](https://tex.z-dn.net/?f=E%28Y%29%20%3D%20%5Cdfrac12%20E%28X%29%20%2B%20%5Cdfrac32%20%5C%5C%5C%5C%20%5Cboxed%7BE%28Y%29%20%3D%202.918%7D)
![E(Y^2) = E\left(\left(\dfrac{X+3}2\right)^2\right) = \dfrac14 E(X^2) + \dfrac32 E(X) + \dfrac94 \\\\ \boxed{E(Y^2) \approx 9.0028}](https://tex.z-dn.net/?f=E%28Y%5E2%29%20%3D%20E%5Cleft%28%5Cleft%28%5Cdfrac%7BX%2B3%7D2%5Cright%29%5E2%5Cright%29%20%3D%20%5Cdfrac14%20E%28X%5E2%29%20%2B%20%5Cdfrac32%20E%28X%29%20%2B%20%5Cdfrac94%20%5C%5C%5C%5C%20%5Cboxed%7BE%28Y%5E2%29%20%5Capprox%209.0028%7D)
Answer:
80 units²
Step-by-step explanation:
Given <u>vertices</u> of a rectangle:
- A = (-8, 5)
- B = (2, 5)
- C = (2, -3)
- D = (-8, -3)
As points A and D share the <u>same x-coordinate</u>, and points B and C share the <u>same x-coordinate</u>, the difference between the different <u>x-coordinates</u> is the length of the rectangle:
![\begin{aligned}\implies \textsf{Length} & = x_B-x_A\\& = 2-(-8)\\& = 2+8\\& = 10\; \sf units\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cimplies%20%5Ctextsf%7BLength%7D%20%26%20%3D%20x_B-x_A%5C%5C%26%20%3D%202-%28-8%29%5C%5C%26%20%3D%202%2B8%5C%5C%26%20%3D%2010%5C%3B%20%5Csf%20units%5Cend%7Baligned%7D)
As points A and B share the <u>same y-coordinate</u>, and points C and D share the <u>same y-coordinate</u>, the difference between the different <u>y-coordinates</u> is the width of the rectangle:
![\begin{aligned}\implies \textsf{Width} & = y_A-y_D\\& = 5-(-3)\\& = 5+3\\& = 8\; \sf units\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cimplies%20%5Ctextsf%7BWidth%7D%20%26%20%3D%20y_A-y_D%5C%5C%26%20%3D%205-%28-3%29%5C%5C%26%20%3D%205%2B3%5C%5C%26%20%3D%208%5C%3B%20%5Csf%20units%5Cend%7Baligned%7D)
Therefore:
![\begin{aligned}\textsf{Area of a rectangle} & = \sf width \times length\\&=8 \times 10\\&=80 \sf \; units^2\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Ctextsf%7BArea%20of%20a%20rectangle%7D%20%26%20%3D%20%5Csf%20width%20%5Ctimes%20length%5C%5C%26%3D8%20%5Ctimes%2010%5C%5C%26%3D80%20%5Csf%20%5C%3B%20units%5E2%5Cend%7Baligned%7D)
Answer:
121.7
Step-by-step explanation:
Use the .746 to round to the nearest decimal place.
To do this you look to the value of the number next to the .7
If it is 5 or higher you round upwards
If it is 4 or lower you round downwards or leave as is.
Therefore .746 rounds to .7
Your answer is 121.7