FORMULAS: Probability and Statistics 1. Summation Notation Sigma means “sum” Ex] 2. Measures of Central Tendency a) Mean - average b) Median – middle number when put in order c) Mode – the number that occurs most often d) Range – difference between highest and lowest 3. Standard Deviation how spread the numerical data is from the mean. S(x) = sample standard deviation = population standard deviation Normal Distribution Probability Permutations – an arrangement of objects in a specific order. In general, the number of permutations of n things, taken n at a time, with r of these things identical repeated is given by ** Combinations – are selections for which ORDER DOES NOT MATTER *** The combination of “n things taken r at a time” is denoted by : 1. A Bernoulli Experiment has only two outcomes (success or failure) Probability = n C r p r q n-r Where n = number of trials r = the number of successes p = the probability of a success q = (1 – p) = the probability of a failure 2. At least r successes in n trials: add together for r, r + 1, r + …, n 3. At most r successes in n trials: add together for 0, 1, …, r Binomial Expansion 1. Use the combination formula (x + y) n = n C 0 x n y 0 + n C 1 x n-1 y 1 + n C 2 x n-2 y 2 + …. + n C n x 0 y n 2. Use Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 etc.
20
Embed
Probability and Statistics - Hamburg Central School · Web viewTo find the roots of a quadratic equation, factor or use the quadratic formula. The Discriminant is b2 – 4ac The Nature
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
FORMULAS:Probability and Statistics
1. Summation Notation Sigma means “sum”
Ex]
2. Measures of Central Tendencya) Mean - averageb) Median – middle number when put in orderc) Mode – the number that occurs most oftend) Range – difference between highest and lowest
3. Standard Deviation how spread the numerical data is from the mean.
S(x) = sample standard deviation
= population standard deviation
Normal Distribution
Probability
Permutations – an arrangement of objects in a specific order.
In general, the number of permutations of n things, taken n at a
time, with r of these things identical repeated is given by
** Combinations – are selections for which ORDER DOES NOT MATTER ***The combination of “n things taken r at a time” is denoted by :
1. A Bernoulli Experiment has only two outcomes (success or failure) Probability = nCr prqn-r
Where n = number of trials r = the number of successes
p = the probability of a successq = (1 – p) = the probability of a failure
2. At least r successes in n trials: add together for r, r + 1, r + …, n
3. At most r successes in n trials: add together for 0, 1, …, r
Standard Form: 2. Given: Find the center and radius.
Center (2, 3) and radius is 3
3. Find the equation of a circle whose endpoints of the diameter are (1, 3) and (7, 5).
[find center]
[find radius: distance from center to one point]
Equation of circle: (x4)2 + (y1)2 = 25 (ans)
Factoring Polynomials
1. Greatest Common Factor
ax + bx = x(a+b)
2. Difference of two perfect squares
a2 – b2 = (a+b)(a-b)
or 9x2 -25 = (3x-5)(3x+5)
3. Factor completely
-means you will need to factor more than once
x3+9x2+14x = x(x2 +9x +14)
= x(x+7)(x+2)
4. Trial and Error
2x2 – x – 6 = (2x + 3)(x – 2)
Multiplying/ Dividing Rational Expressions1. If dividing, take reciprocal of 2nd fraction and change to multiplication2. Factor completely3. Cross cancel4. Multiply across
–2 (answer)
Adding/ Subtracting Rational Expressions1. Find the least common denominator (give missing part to top AND bottom of each fraction)2. Add or subtract numerators, (keep the denominator!)3. Simplify (if necessary)
Example:
(answer)
Simplifying Complex Fractions1. Find the common denominator2. Multiply each part by the common denominator3. Simplify
Example:
3–x (answer)
Solving Fractional Equations1. Find the common denominator2. Multiply each part by the common denominator – this should eliminate all fractions3. Solve the new equation4. Check
Relations and Functions1. A relation is any set of ordered pairs.
2. A function is a relation in which every element in the
domain corresponds to only one element in the range.
(Vertical line test to see if we have a function)
3. The domain is the set of first elements (x-value)
4. The range is the set of second elements (Y-values)
Special Relations and Functions1. Circle ax2 + ay2 = c
two squared terms
same coefficient and sign
2. Ellipse ax2 + by2 = c
two squared terms
different coefficient
same sign
3. Hyperbola
Case 1: ax2 -by2 = c
Two squared terms
Different signs
Case 2: xy = c
Aka. Inverse Variation
5. Vertical Parabola y = ax2 + bx + c
axis of symmetry:
when a > 0
when a < 0
6. Horizontal Parabola: x =ay2 + by + c
when a > 0
axis of symmetry y =
when a < 0
6
Variation1. Direct Variation y = xc ( Graph is a straight line)
2. Inverse Variation xy = c (Graph is a hyperbola)
Other Functions1. Composition of Functions
**Performed right to left
2. Inverse Function f-1 (x)
Rule: Switch the x and y and solve for y. This is a reflection over
the line y = x.
Scientific Notation
1. 38.7 = 3.87 x 101
2. .0387 = 3.87 x 10-2
Exponent Laws1. (xa)(xb) = xa+b
2. xab+c = (xab) (xc)
3.
4. (xa)b = xab
5. (xyz)a = xayaza or (xbyc)a = xbayca
6. =
7. x0 = 1, (x 0)
8. x-a = or = xa
9. =
10. =
11. (xy)m = xmym
12. xy-1 =
13.
14. (xy)-1 =
15. x-1 + y-1 =
Exponential Functions1. y = ax
2. y =
Solving Equations with Fractional Exponents1. Isolate the variable with the fractional exponent
2. Raise both sides to the reciprocal exponent
3. Check.
Example:
Logarithms
1. y = logbx x=by
2. Log Laws:
Product: logb(AB) = logbA + logbB
7
Quotient logb = logbA - logbB
Power logb = clogbA
Caution: There are no laws for addition or subtraction,
(ex: logb(A + B) can’t be performed! but logbA + logbB = lobb AB)
3. Solve for x: log2x + log2(x2) = 3
log2 x(x2) = 3 [condense]
23 = x(x2) [exponential form]
8 = x2 2x [solve]
0 = x2 2x 8
0 = (x4)(x+2)
x = 4 and x = 2(reject, can’t have negative logs)
ans: x = 4
4. Solve for x: log (x+3) log x = log 4
[condense]
4x = x+3 [cross multiply]
3x = 3 [solve]
ans: x = 1
example: y = log2 x
Exponential functions and logarithms are inverses
Ex: If y = 2x then f-1 is x = 2y and in log form y = log2 x
Since they are inverses they are ry=x
Solving Exponential Equations Using Logs1. Isolate the base raided to the variable.
2. Take the common log of both sides
Example:
Change of Base for Logs
Exponent Phrases you Should know!
1. A logarithm IS an exponent.
2. Fractional exponents represent ROOTS.
Ex.
3. Negative exponents represent FRACTIONAL
EXPRESSIONS. Ex. x-2 =
8
cb
a
Trigonometry1. Given:
2. Standard Position Angle – The initial ray is on the positive x-axis with the vertex at the origin and the terminal ray anywhere on the Cartesian graph.
3. Positive Angles open counter-clockwise: negative angles open clockwise.
= 135
= -45
4. Co-terminal Angles are angles that have the same terminal rays Ex: co-terminal
5. Trig ratios of co-terminal angles are equal. Ex: tan = tan ( )
Special Right Triangles
1. The Right Triangle Leg opposite = half the hypotenuse Leg opposite = half the hypotenuse times
3
1
2
60
30
45
452
1
1
2. The Right Triangle Leg = half the hypotenuse times Hypotenuse = leg times
Trig Reference Table
1. Table of Positive Trig Functions
all 6
sec
cos
cot
tan
csc
s in
2. Quotient Identities
3. Reciprocal Identities
4. Pythagorean Identities
5. Sum of Two Angles 6. Difference of Two Anglessin(A + B) = sinA cosB + cosA sinB sin(A – B) = sinA cosB – cosA sinBcos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB
tan(A + B) = tan(A B) =
7. Double angles
9
8. Half angles
10
s
r
The Unit Circle
(0,-1)
(-1,0)
(0,1)
(1,0)
(x,y) = (cos, sin)
Inverse Trig
Arcsin means angle whose sine is
Ex] x = arc cos cos x =
Ex] x = arc sin
Use 45 as a reference angle
Radians1. A radian is a measure of an angle that intercepts an arc equal to the length of the radius of the circle.
Central Angle (in radians) =
Cofunctions CO is abbreviation for COmplementary.
EX]
Ex]
Ex] sin(x + 10) = cos (3x) x + 10 + 3x = 90
x = 20
Solving Trig Equations
Ex] Find all for
2sin2 + 5 sin = 32sin2 + 5 sin -3 = 0
Degree 30 60 45 0 90 180 270 360
radian 0
sin 0 1 0 -1 0
cos 1 0 -1 0 1
tan 1 0 dne 0 dne 0
11
Trig Graphs
1. y = a sin b , y = a cos bAmplitude - = maximum vertical height
The amplitude for tangent is undefined
Frequency = = # of full cycles from 0 to
Period = = length of one full cycle
2. Basic Graphs a) y = sin
b) y = cos
c) y = tan
3. If you need to write the equation of a trig function by looking at the graph, use the following formulas:
choose either a) y = a cos(bxh) + c or b) y = a sin(bxh) + c
b = frequency = the number of times the curve repeats from 0 to
h = horizontal shift
Trig Applications
1. Law of Cosines (SAS or SSS given) – Any side squared equals the sum of the squares of the other two sides minus two times the product of the other two sides times the cosine of the angle between those two sides.
C
B
A
c
b
a
a2 = b2 + c2 – 2bc cosA
2. Law of Sines ( ASA or AAS given) – Any side ratioed with the sine of the angle opposite it equals any side ratioed with the sine of the angle opposite it.
3. Area of a Triangle – The area of any triangle is equal to one half the product of any tow sides times the sine of the angle between those two sides.
4. Area of a Parallelogram K = ab sin C
5. The Ambiguous Case (SSA given)
There may be 0, 1, or 2 possible triangles. Solve for the unknown angle using the law of Sines and also see if the obtuse angle from Quadrant II (use the solved for angle as a reference angle) is a viable answer.
6. Forces
resultant force1st force
2nd force
remember: opposite sides of a parallelogram are consecutive angles are supplementary