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BIO 440 – Wildlife Ecology
Mathematics Review
Functions: Given 2 variables x and y, if for each x there is a corresponding value y, then we say
that y is a function of x, i.e. y = f(x).
We will deal with 4 common types of functions:
1. Linear (general form of y = a + bx, where a = intercept, b = slope)
2. Polynomial (general form of y = ao + a1x + a2x2 + ... + an-1xn-1 + anxn)
Note that a linear function is a special (i.e., 1st degree) polynomial function.
3. Exponential (general form y = ex, where e = 2.7183 is the base of natural [Naperian]
logarithms)
4. Power (general form of y = xn)
Powers and Logarithms: For the expression an, we refer to a as the base and n as the exponent.
Some algebraic properties of a base raised to a power are...
1. If a common base: an x am = an+m; also an/am = an-m
2. If a common exponent: an x bn = (ab)n; also an/bn = (a/b)n
3. Double exponentiation:
a 
n m
 an m
For logarithms of base a, loga(an) = n.
Some algebraic properties of logarithms (note parallels with properties for exponents)...
1. loga  m  n   loga m+loga n
2.
2. loga(m/n) = logam - logan
3. log a m n =n  log a m
4. Relation of different bases: logan = logbn x logab
5. By definition, loga1 = 0 and logaa = 1.
BIO 440 – Wildlife Ecology
Miscellaneous: The following notation often is useful in summarizing expressions...
1. Sigma notation: the sum of a series of n numbers x1, x2, ... ,xn can be expressed as
n
 x , where xi represents the ith member of the series.
i
i=1
3
Example: For x1 = 3, x2 = 7, and x3 = 26,
 x =36.
i
i=1
2. Pi notation: the product of a series of n numbers x1, x2, ... ,xn can be expressed as
n
 x , where xi represents the ith member of the series.
i
i 1
3
Example: For x1 = 3, x2 = 7, and x3 = 2,
 x =42.
i
i=1
3. Factorials: n! = (n)(n-1)(n-2)...(n-[n-2])(1)
Example: 4! = (4)(3)(2)(1) = 24
For n>m, n!/m! = (n)(n-1)(n-2)...(n-m+1)
Example: 4!/2! = (4)(3) = 12
n
n
4. Combinatorials: n C x   x  = the number of combinations of
n things taken x at a time =
n!
.
 n-x  !x!
Identity: 0! = 1
And finally, a potpourri of useful tricks and trivia:
5. Special product (the FOIL principle)
(a + b)2 = (a + b)(a + b) = First + Outside + Inside + Last
= a2 + ab + ab + b2
= a2 + 2ab + b2
6. Quadratic formula: If ax2 + bx + c = 0, a  0, then
1
x=
-b   b 2 -4ac  2
2a
BIO 440 – Wildlife Ecology
7. Radicals:
n
a a
m
m
n
8. Reciprocal of a power: a-m = 1/am
Rules of Differentiation: The following rules will come in handy from time to time.
1.
dc
 c'=0, where c = a constant.
dx
2.
dx
 y'=1
dx
3.
d  cu 
d  2x 

du 
du
dx
 c  ex: c=2, u=x;
2
2
 2
dx
dx 
dx
dx
dx

2


d  3x+1
du n
1 du
1
n-1 du
4.
 nu
 2  3x+1
 2  3x+1  3  6 3x+1 
ex: u=3x+1, n=2;
dx
dx 
dx
dx

5.
d  u+v  du dv 
d  u+v  d  2x+3 d  2x  d  3


 ex: u=2x, v=3;



 2+0=2 
dx
dx dx 
dx
dx
dx
dx

d  uv  d  3x  x 
d  3x  

dx
d  uv 
dv
du ex: u=3x, v=x;

 3x
x

 u  v
6.
dx
dx
dx
dx 

dx
dx
dx
  3x 1+x  3=6x

7.
u
d   v  du  u  dv
 v   dx
dx
2
dx
v
d  2x+1 
dx
u 
d    2x+1  x 

dx
dx 
v 


ex: u = x, v = 2x + 1;
2
dx
 2x+1
 2x+1 1-x  2
1


2
2
 2x+1
 2x+1
BIO 440 – Wildlife Ecology
8.
d  eu 

d  e3x 
du 
 e  ex: u=3x;
 e3x  3
dx
dx 
dx


u
du
d  ln u  dx

9.
dx
u
d  ln 7  7 1 


 
ex: u=7x;
dx
7x x 

dy
d2y
, is simply the derivative of
(i.e., the
2
dx
dx
derivative of the derivative). It is interpreted as the rate of change of the rate of change (often
referred to as acceleration).
Note that the second derivative of f(x), termed
Integration of One Variable:
The name of the integration game is, given f'(x), to find F(x), i.e., the function which was
differentiated to get f'(x).
That is, find F(x) such that
d  F x 
dx
 f  x  . Equivalently, d(F(x)) = f(x)dx = F(x) + C, where C
is the constant of integration.
Example: Let f'(x) = 3x2. F(x) = x3 is a perfectly good answer, because when
differentiated, it gives 3x2. Note that x3 + 10, x3 + 99, etc. also work.
Some rules of integration...
1.
 dx=x+C
2.
 kdx=k  dx=kx+C
3.
 f  x   f  x  dx=  f  x dx+ f  x  dx
1
2
1
 u n+1 
4.  u du= 
  C, for all n  1
 n+1 
n
1
dx     du=ln u + C
u
5.
u
6.
 e du  e
-1
u
u
+C
2
BIO 440 – Wildlife Ecology
Example: Evaluate
e
4x
e
4x
dx . Let u = 4x, then du = 4dx, so dx =
du
. Thus
4
 du   1 
1
1
dx=  eu       e u du    e u +C=   e 4x +C
 4  4
4
4
 udv=uv- vdu
Example: Evaluate  xe dx . Let u = x, v = ex, then du = 1 and dv = ex.
Thus  xe dx=xe - e 1dx=xe -e +C.
7. Integration by parts:
x
x
x
x
x
x
Definite (Riemann) Integrals: For an integrable function f(x),
the integral of f(x) evaluated at x = a, minus the integral of f(x) evaluated at x = b, is
denoted by
a
 f  x dx=F  a   F  b   F  x  
a
b
b
(note that the constant of integration is set equal to zero)
1
Example:
 1  4x 14  1  1 0
4x
e
dx=
  e  0     e -e   .43
0
4
4
4
Definite integrals are useful in a variety of situations, as, e.g., determining the total number
of animals occupying an area over a given time span (i.e., animal-days of occupancy).
BIO 440 – Wildlife Ecology
Statistics Review
1. Measures of Central Tendency
A. Arithmetic mean: For a population, the mean is designated by  or E(X), i.e.,
the expectation of random variable X. For a sample from a population,
n
xi
the sample mean, x , is computed as x =  , where n represents the total number
n
of observations and xi is the ith observation.
B. Median: the middle measurement (50th percentile) in a set of data.
median = x(n+1)/2 if n is odd,
= (xn/2 + x(n/2)+1)/2 if n is even.
2. Measures of Variability
A. Variance: the average squared deviation of observations about their mean. The
variance of a population is designated by 2 or Var(X). For a sample from a
population, the sample variance, s2, is computed as


2

x i -x 



s2 = 
 n-1
An easier computational form of the sample variance is given by
 x 2 -  x 2 
 i  i 
n


.
s2 = 
n-1
 
B. Standard deviation: the standard deviation is merely the square root of the
variance. It expresses variability in the original units of measurement. Population
standard deviation = , and sample standard deviation = s.
3. Standard Normal Deviates
It often is desirable to express data as deviations from a standard normal distribution (i.e, a
Normal distribution with  = 0 and  = 1). This is accomplished as follows:
standardized deviate =
 x-x 
s
BIO 440 – Wildlife Ecology
Probability Distributions: We will deal only with a few common probability distributions...
n!
n-x
 p x 1-p  , where P(X=x) is the probability of x
 n-x !x!
"successes" or occurrences in n independent (Bernoulli) trials, with p being the probability
of success in each Bernoulli (coin flip) trial and 1-p = q being the probability of failure.
Mean = np
Variance = npq
1. Binomial: P  X=x  
2. Poisson: P  X=x  
 x  e- 
x!
.
Mean = Variance = 
3. Normal: P  X=x  
-  x- 
exp
, where  is the population mean,  is the
2 2
2
1
1
  2  2
population standard deviation, and exp[x] = ex.
4. Rectangular (continuous uniform): This is the probability distribution which generates
random dispersion patterns. Imagine a line segment starting at a and ending at a + b. In
other words, if our range spans the distance b, then the probability of occurrence is equal for
each point across the interval [a,a+b]. Formally,
P(X=x) = 1/b, with Mean = (a + b)/2 and Variance = b2/12.
5. We also will use several distributions for testing hypotheses. These include the Normal
(above), Chi-squared, F, and Student's T distributions. Some common tests are described
below.
A. One-sample t test with n-1 degrees of freedom (df):
t=
x  
 s 


 n
B. Two-sample t test with n1 + n2 - 2 df:
t=
x
1
 x2
s x1 x 2
 , where
BIO 440 – Wildlife Ecology
1
s x1 x 2
 s p 2  s p 2  2
 
 , and

 n1  n 2 


2
sp = 
 x
1i
-x1
   x
2
 n1 +n 2 -2 
2i

2
-x 2 

C. Chi-square goodness-of-fit test with k-1 df:
  fi -Fi 2 
 = 
 , where fi and Fi are observed and expected frequencies for
F
i=1 

i

category i (k categories in all).
k
2