Answer:
![Sd(X) =\sqrt{1.2}=1.095](https://tex.z-dn.net/?f=%20Sd%28X%29%20%3D%5Csqrt%7B1.2%7D%3D1.095)
Step-by-step explanation:
Previous concepts
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Solution to the problem
Let X the random variable of interest, on this case we now that:
![X \sim Binom(n=5, p=0.4)](https://tex.z-dn.net/?f=X%20%5Csim%20Binom%28n%3D5%2C%20p%3D0.4%29)
The probability mass function for the Binomial distribution is given as:
![P(X)=(nCx)(p)^x (1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%29%3D%28nCx%29%28p%29%5Ex%20%281-p%29%5E%7Bn-x%7D)
Where (nCx) means combinatory and it's given by this formula:
![nCx=\frac{n!}{(n-x)! x!}](https://tex.z-dn.net/?f=nCx%3D%5Cfrac%7Bn%21%7D%7B%28n-x%29%21%20x%21%7D)
The mean for the binomial distribution is given by:
![E(X) =np=5*0.4=2](https://tex.z-dn.net/?f=%20E%28X%29%20%3Dnp%3D5%2A0.4%3D2)
And the variance is given by:
![Var(X) = np(1-p) =5*0.4*(1-0.4)=1.2](https://tex.z-dn.net/?f=Var%28X%29%20%3D%20np%281-p%29%20%3D5%2A0.4%2A%281-0.4%29%3D1.2)
And the deviation is just the square root of the variance so we got:
![Sd(X) =\sqrt{1.2}=1.095](https://tex.z-dn.net/?f=%20Sd%28X%29%20%3D%5Csqrt%7B1.2%7D%3D1.095)
Answer:
Step-by-step explanation:
Answer:
General Formulas and Concepts:
<u>Pre-Calculus</u>
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Integration
- Integrals
- Definite/Indefinite Integrals
- Integration Constant C
Integration Rule [Reverse Power Rule]: ![\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bx%5En%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bx%5E%7Bn%20%2B%201%7D%7D%7Bn%20%2B%201%7D%20%2B%20C)
Integration Rule [Fundamental Theorem of Calculus 1]: ![\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Eb_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20F%28b%29%20-%20F%28a%29)
U-Substitution
- Trigonometric Substitution
Reduction Formula: ![\displaystyle \int {cos^n(x)} \, dx = \frac{n - 1}{n}\int {cos^{n - 2}(x)} \, dx + \frac{cos^{n - 1}(x)sin(x)}{n}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bcos%5En%28x%29%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bn%20-%201%7D%7Bn%7D%5Cint%20%7Bcos%5E%7Bn%20-%202%7D%28x%29%7D%20%5C%2C%20dx%20%2B%20%5Cfrac%7Bcos%5E%7Bn%20-%201%7D%28x%29sin%28x%29%7D%7Bn%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx)
<u>Step 2: Integrate Pt. 1</u>
<em>Identify variables for u-substitution (trigonometric substitution).</em>
- Set <em>u</em>:
![\displaystyle x = sin(u)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%20%3D%20sin%28u%29)
- [<em>u</em>] Differentiate [Trigonometric Differentiation]:
![\displaystyle dx = cos(u) \ du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20dx%20%3D%20cos%28u%29%20%5C%20du)
- Rewrite <em>u</em>:
![\displaystyle u = arcsin(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20u%20%3D%20arcsin%28x%29)
<u>Step 3: Integrate Pt. 2</u>
- [Integral] Trigonometric Substitution:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \int\limits^a_b {cos(u)[1 - sin^2(u)]^\Big{\frac{3}{2}} \, du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%5Ea_b%20%7Bcos%28u%29%5B1%20-%20sin%5E2%28u%29%5D%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%5C%2C%20du)
- [Integrand] Rewrite:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \int\limits^a_b {cos(u)[cos^2(u)]^\Big{\frac{3}{2}} \, du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%5Ea_b%20%7Bcos%28u%29%5Bcos%5E2%28u%29%5D%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%5C%2C%20du)
- [Integrand] Simplify:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \int\limits^a_b {cos^4(u)} \, du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cint%5Climits%5Ea_b%20%7Bcos%5E4%28u%29%7D%20%5C%2C%20du)
- [Integral] Reduction Formula:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{4 - 1}{4}\int \limits^a_b {cos^{4 - 2}(x)} \, dx + \frac{cos^{4 - 1}(u)sin(u)}{4} \bigg| \limits^a_b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B4%20-%201%7D%7B4%7D%5Cint%20%5Climits%5Ea_b%20%7Bcos%5E%7B4%20-%202%7D%28x%29%7D%20%5C%2C%20dx%20%2B%20%5Cfrac%7Bcos%5E%7B4%20-%201%7D%28u%29sin%28u%29%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b)
- [Integral] Simplify:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(u)sin(u)}{4} \bigg| \limits^a_b + \frac{3}{4}\int\limits^a_b {cos^2(u)} \, du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bcos%5E3%28u%29sin%28u%29%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3%7D%7B4%7D%5Cint%5Climits%5Ea_b%20%7Bcos%5E2%28u%29%7D%20%5C%2C%20du)
- [Integral] Reduction Formula:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(u)sin(u)}{4} \bigg|\limits^a_b + \frac{3}{4} \bigg[ \frac{2 - 1}{2}\int\limits^a_b {cos^{2 - 2}(u)} \, du + \frac{cos^{2 - 1}(u)sin(u)}{2} \bigg| \limits^a_b \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bcos%5E3%28u%29sin%28u%29%7D%7B4%7D%20%5Cbigg%7C%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3%7D%7B4%7D%20%5Cbigg%5B%20%5Cfrac%7B2%20-%201%7D%7B2%7D%5Cint%5Climits%5Ea_b%20%7Bcos%5E%7B2%20-%202%7D%28u%29%7D%20%5C%2C%20du%20%2B%20%5Cfrac%7Bcos%5E%7B2%20-%201%7D%28u%29sin%28u%29%7D%7B2%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%5Cbigg%5D)
- [Integral] Simplify:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(u)sin(u)}{4} \bigg| \limits^a_b + \frac{3}{4} \bigg[ \frac{1}{2}\int\limits^a_b {} \, du + \frac{cos(u)sin(u)}{2} \bigg| \limits^a_b \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bcos%5E3%28u%29sin%28u%29%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3%7D%7B4%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B2%7D%5Cint%5Climits%5Ea_b%20%7B%7D%20%5C%2C%20du%20%2B%20%5Cfrac%7Bcos%28u%29sin%28u%29%7D%7B2%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%5Cbigg%5D)
- [Integral] Reverse Power Rule:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(u)sin(u)}{4} \bigg| \limits^a_b + \frac{3}{4} \bigg[ \frac{1}{2}(u) \bigg| \limits^a_b + \frac{cos(u)sin(u)}{2} \bigg| \limits^a_b \bigg]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bcos%5E3%28u%29sin%28u%29%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3%7D%7B4%7D%20%5Cbigg%5B%20%5Cfrac%7B1%7D%7B2%7D%28u%29%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7Bcos%28u%29sin%28u%29%7D%7B2%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%5Cbigg%5D)
- Simplify:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(u)sin(u)}{4} \bigg| \limits^a_b + \frac{3cos(u)sin(u)}{8} \bigg| \limits^a_b + \frac{3}{8}(u) \bigg| \limits^a_b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bcos%5E3%28u%29sin%28u%29%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3cos%28u%29sin%28u%29%7D%7B8%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3%7D%7B8%7D%28u%29%20%5Cbigg%7C%20%5Climits%5Ea_b)
- Back-Substitute:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(arcsin(x))sin(arcsin(x))}{4} \bigg| \limits^a_b + \frac{3cos(arcsin(x))sin(arcsin(x))}{8} \bigg| \limits^a_b + \frac{3}{8}(arcsin(x)) \bigg| \limits^a_b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bcos%5E3%28arcsin%28x%29%29sin%28arcsin%28x%29%29%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3cos%28arcsin%28x%29%29sin%28arcsin%28x%29%29%7D%7B8%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3%7D%7B8%7D%28arcsin%28x%29%29%20%5Cbigg%7C%20%5Climits%5Ea_b)
- Simplify:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{3arcsin(x)}{8} \bigg| \limits^a_b + \frac{x(1 - x^2)^\Big{\frac{3}{2}}}{4} \bigg| \limits^a_b + \frac{3x\sqrt{1 - x^2}}{8} \bigg| \limits^a_b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B3arcsin%28x%29%7D%7B8%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7Bx%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B4%7D%20%5Cbigg%7C%20%5Climits%5Ea_b%20%2B%20%5Cfrac%7B3x%5Csqrt%7B1%20-%20x%5E2%7D%7D%7B8%7D%20%5Cbigg%7C%20%5Climits%5Ea_b)
- Rewrite:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{3arcsin(x) + 2x(1 - x^2)^\Big{\frac{3}{2}} + 3x\sqrt{1 - x^2}}{8} \bigg| \limits^a_b](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B3arcsin%28x%29%20%2B%202x%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%2B%203x%5Csqrt%7B1%20-%20x%5E2%7D%7D%7B8%7D%20%5Cbigg%7C%20%5Climits%5Ea_b)
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
![\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{3arcsin(a) + 2a(1 - a^2)^\Big{\frac{3}{2}} + 3a\sqrt{1 - a^2}}{8} - \frac{3arcsin(b) + 2b(1 - b^2)^\Big{\frac{3}{2}} + 3b\sqrt{1 - b^2}}{8}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Ea_b%20%7B%281%20-%20x%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B3arcsin%28a%29%20%2B%202a%281%20-%20a%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%2B%203a%5Csqrt%7B1%20-%20a%5E2%7D%7D%7B8%7D%20-%20%5Cfrac%7B3arcsin%28b%29%20%2B%202b%281%20-%20b%5E2%29%5E%5CBig%7B%5Cfrac%7B3%7D%7B2%7D%7D%20%2B%203b%5Csqrt%7B1%20-%20b%5E2%7D%7D%7B8%7D)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Book: College Calculus 10e
Drop a perpendicular from an obtuse vertex.
That makes a 30 - 60º right triangle, with
hypotenuse 6 and short leg 3. So the long leg,
which is the height of the rhombus, is 3√3 .
Since a rhombus is a parallelogram, its area
is base x height = 6 x 3√3 = 18√3 .