Statistics Flashcards
(101 cards)
1
Q
F (x) = f/n
A
Relative frequency = frequency/ total number of data
2
Q
CDF plot
A
Fx against data point
3
Q
What does the gradient of a CDF plot mean?
A
The steepest part of CDF plot corresponds to most number of data points. Shallower CDF -> low number of data points
4
Q
Skewness?
A
Which way the curve leans how asymmetrical it is
5
Q
Positive skew
A
Skews to the left
6
Q
Negative skew
A
Skews right
7
Q
Interquartile range?
A
Difference between third quartile and first quartile
8
Q
Sample space?
A
The set Ω of all possible outcomes of an experiment
9
Q
Event?
A
A sub-set A of the sample space A c Ω
10
Q
Α n B?
A
A and B occurring together (intersection)
11
Q
A u B
A
A or B occurring (union)
12
Q
_
A?
A
Not A (complement)
13
Q
Mutually exclusive?
A
Events cannot occur at same time
14
Q
Independent?
A
Event is not affected by previous events
15
Q
Dependent?
A
Event is affected by other event
16
Q
Permutations?
A
Order of outcome is important
17
Q
Combinations?
A
Order of outcomes is not important
18
Q
P(A|B)?
A
Probability of A given that B has already occurred
19
Q
P(A|B)P(B) = P(B|A)P(A) = ?
A
P(A n B)
20
Q
What does it mean if P(A|B) = P(A)?
A
Events A and B are independent
21
Q
Partitioned sample space?
A
The events are non-empty, non- overlapping whose union forms the whole sample space
22
Q
ΣP(B|A)P(A) = ? (Total probability law)
A
P(B)
23
Q
P(A|B)P(A) / P(B) = ? (bayes law)
A
P(A|B)
24
Q
P(A|B n C) P(B|C) P(C) = ?
A
P(A n B n C)
25
Sx^2 = 1/(n-1) Σ(xi-x~)^2
Sample variance
Sample deviation = Sx
N= number of data points
x= value (each one)
x~= mean
26
Vx= Sx/ x~
Coefficient of variation = standard deviation/ mean
27
dx=1/n * Σ|xi-x~|
Mean absolute deviation = dx
n= number of data points
xi = data points
x~= mean
28
Unbiased skewness= sqrt(n-1)/ (sqrt(n)*(n-2)(σ^3) * Σ(xi-x~)^3
n= number of data points
σ=standard deviation
xi= data point
x~ = mean
29
Biased skewness = (1/n) * Σ(xi-x~)^3 / σ^3
n= number of data points
xi= data point
x~ = mean
σ= standard deviation
30
What order is mode median and mean for positive skewness?
Mode - median - mean
31
In what order is mode median and mean for a symmetric curve?
All equal
32
In what order is mode median and mean for negative skewness?
Mean-> median -> mode
33
cov= 1/(n-1) Σ(xi-x~)(yi-y~)
Sample convergence
n= number of data points
xi= each x point
x~ = mean x
yi= y point
y~ = y mean
34
Cxy =1/(n-1) Σ(xi-x~)(yi-y~) / SxSy
Cxy= sample correlation coefficient (has to be between -1 and 1)
n= number of points
xi= x point
x~ = x mean
yi= y point
y~ = y mean
Sx= x standard deviation
Sy= y standard deviation
35
What does Cxy= -1 mean?
Perfect negative linear correlation
36
What does Cxy= 0 mean?
No correlation
37
What Cxy= 1 mean?
Perfect positive linear correlation
38
Variable?
A quantity that can vary
39
Random?
The result will be the outcome of a random experiment
40
Discrete?
Has a limited number of outcomes
41
Continuous?
Has a limitless number of outcomes in between two points
42
PDF=f(x) = dF(x)/ dx
Probability density function = the gradient of CDF plot
43
What is the area under a CDF plot?
1
44
The steepest part of a CDF plot is where on a PDF (probability density function) plot?
The peak
45
Cov(X,Y)
= E[(X-E[X]) (Y-E[Y]))
= E[XY]-E[X]E[Y]
46
Cov (X,X)?
Var (X)
47
Var(X+Y)?
Var (X) + Var(Y) + 2Cov(X,Y)
48
What is the convenience of random variables X and Y?
A measure of the linear dependence between these variables
49
Cov(cx,Y) = Cov(X,cY)
c*Cov(X,Y)
50
Cov(X+Y,Z) ?
Cov(X,Z) + Cov(Y,Z)
51
Cov (X,Y) =
(If symmetrical???)
Cov(Y,X)
52
Cov(X,c)?
0
53
Cov(Σ ai Xi, Σbj Yj) =
Σ Σ ai bj Cov(Xi,Yj)
54
How do you standardise a value for confidence intervals to N(0,1)
Z-E(X) / σ
55
When do you apply a continuity correction?
When you are using a normal distribution to approximate a discrete distribution (binomial, poisson)
56
What is the continuity correction?
+/- 0.5 to values
57
Sqrt (n)= (z*σ)/ E
n = minimum sample size
z = x value relating to probability
E= margin of error
σ = standard deviation
58
_
X - μ / σ/ sqrt(n) ~ N(0,1)
X = value comparing to
μ = mean
σ = standard deviation
n = population
59
Cis for the mean with known standard deviation?
P(μ- zσ/sqrt(n) < X < μ + zσ/sqrt(n) ) = 1- α
60
Cis for the mean with unknown standard deviation n?
_
X +- t*s/sqrt(n) = CI
Where v (degree of freedom) for t is n-1
61
Normal approximation for large sample sizes of binomial?
N(p, p(1-p)/n)
62
Inf-0 Σ μj/ j!
e^μ
63
pXY = Cov(X,Y) / sqrt(Var(X) Var(Y))
Correlation coefficient = covariance of X and Y / standard deviation of x and y
64
-1 < p(X,Y) < 1
Correlation coefficient is between -1 and 1
65
If p(X,Y) = 0
X and Y are uncorrelated
66
p(X,Y) > 0
X and Y are positively correlated
67
p(X,Y) < 0
X and Y are negatively correlated
68
FXY (j,k) = P( X
Joint cumulative function
69
Inf to -inf Σ Σ fXY (x,y) dxdy = 1
Sum of all fXY over infinity =1
70
Σy -inf Σ x -inf fXY (u,v) dudv = FXY (j,k)
Sum of all fxy up to x and y is the cumulative function
71
Px (j) = Σ PXY (j,k) from 0 to inf
-
72
fx(x) = Σ fXY (x,y) dy from inf to -inf
-
73
If X and Y are independent?
- PXY (j,k) = PX(j) PY(k) (discrete case)
- fXY (x,y) = fX(x) fY(y) (continuous case)
- fX|Y (x|y) = fX(x)
74
fX|Y(x|y) = fXY (x,y) / fY(y)
-
75
Px|Y (j,k) = P(X= j and Y=k) / P(Y=k)
-
76
What is simple linear regression?
A process where you try and find the line of best fit which is the line that has the minimum
Σ (yi-a-bxi)^2
77
b = Cov xy/ var x
B for line of best fit which
78
a= ymean- b*xmean
A for line of best fit
79
r2 = var (a+bx) / vary
Coefficient of determination = variance from the line of best fit to data points / variance of y
80
r^2 = cov^2xy / varx * vary
Coefficient of determination
81
Z ~ N(0,σ^2/Z)
The errors Zi = y-a-bx can be estimated as independent and normally distributed
82
Yi|X = xi ~ N(a+bxi, σ^2Z)
Assumption
83
a ~ (a, (1/n + mx^2/nvarx) Varz)
Normal distribution of a
84
Var(a) = (1/n + mx^2/ nvarx) VarZ
a= value for y= a+ xb
n= population
mx = mean of x
varx = variance of x
Var Z = variance of error values
85
VarZ = 1/n-2 (nvary- ncov^2xy/ varx)
VarZ = variance of Z errors
n= population
Vary = variance of y
Cov= covariants of x and y
varx = variance of x
86
b ~ N(b, VarZ/ nvarx)
b can be normally distributed for y= a+xib
87
Cis for b
b +/- sqrt(Varb) * z
88
What t should you use for no known population of linear regression error z?
n-2
89
Cis for a?
a +- sqrt(Vara) * z
90
Cis for a+bx0
a + bx0 +- (t sqrt var (a+bx0))
Var (a+bx0) = Var(Z) (1/n + (x0-mx)^2/nvarx)
91
CIS got a+bx0 + Z
Add + 1 to 1/n for formula for a+bx0
92
Test on the slope parameter b?
1. Null hypothesis H0: b=b*
2. Alternative Hypothesis H1: b=/ b*
3. T=2.953 lt b-b/ sqrt(VarZ/nvarx)
4. Distribution of test statistic T~t(n-2)
5. Critical region R |T| > t n-2
6. Evaluate T under H0. If |t0|> tn-2 Reject H0
7. p-value p|T|>t0
93
Test on the correlation for linear regression?
1. Null hypothesis H0: corr(X,Y) =0
2. Alternative Hypothesis H1: Corr(X,Y) =/0
3. Test statistic V= sqrt(n-3)/2 ln((1+cxy)(1-corrXY)/(1-cxy)(1+corrXY))
4. Distribution of test statistic V~N(0,1)
5. Critical region |V|> z
6. Evaluate V under H0 if |v0|>z -> Reject H0
7. p-value p(|V|>v0)
94
ssreg = nvar(a+bx)
Regression sum of squares
95
ssrcs = (n-2) VarZ
Residual sum of squares
96
sstot = ssreg + ssrcs = nvary
Total sum of squares = regression sum of squares + residual sum of squares
97
F= ssreg/1 / ssrcs/n-2 ~ F(1,n-2)
F distribution
98
Test on the regression F-test for linear regression?
1. Null hypothesis H0: b=0
2. Alternative hypothesis H1: b=/0
3. Test statistic F= ssreg/ssrcs/n-2
4. Distribution of test statistic F~F(1,n-2)
5. Critical region F>f1,n-2
6. Evaluate F under H0 if f0>f1,n-2 -> Reject H0
7. p-value p(F>f0)
99
Mean exponential?
1/λ
100
Log - likelihood function?
l(λ) = -λ Σti + n lnλ
101
Variance of two parts together?
Var Z = VarX + VarY + 2 σ x σ y
