Stats Flashcards

1
Q

Binomial distribution conditions

A
  • A finite number of trials are carried out
  • The trials are independent
  • The outcome of each trial is either a success or a failure
  • The probability of a successful outcome is the same for each trial
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2
Q

Equation for binomial distribution

A

P(X=r)=nCrq^(n-r)p^r

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3
Q

Expectation and variance for binomial distribution

A

E(x)=np

Var(X)=npq

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4
Q

Mode of binomial distribution

A

If p=0.5 and n is odd, there are two modes.

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5
Q

Type I error

A

When H0 is rejected when it is true

P=significance level

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6
Q

Type II error

A

When H0 is accepted but it is false

P=Probability of H0 being accepted under conditions specified by H1

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7
Q

Least squares regression line of y on x

A

Minimises square of vertical distance m
y= a + bx
b = sxy/sxx
Used when x is independent variable and or you want to predict the value of y

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8
Q

What is the difference between interpolating and extrapolating

A

Using the regression line to make predictions within the range of data is interpolating, going outside of the range is extrapolating and is unreliable

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9
Q

What is a pdf

A

It is a function that allocates probability to each discrete value of X or it allocates probability to areas for a continuous variable

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10
Q

E(g(X))

A

Sum g(x)*P(X=x)

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11
Q

Expectation rules

A

E(kX) = kE(X)
E(k) = k
E(kX +c) = kE(X) + c
E(f(X) + g(X)) = E(f(X)) + E(g(X))

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12
Q

Variance rules

A
Var(X) = E(X^2) + E^2(X)
Var(k) = 0
Var(kX) = k^2Var(X)
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13
Q

CDF

A

Cumulative distribution function

F(x) = P(X=< x)

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14
Q

Multiple independent random variables

A
E(X+Y) = E(X) + E(Y)
E(X-Y) = E(X) - E(Y)
E(X1+X2+…+Xn) = nE(X)
Variance rules require independence
Var(X+-Y) = Var(X) + Var(Y)
Var(X1+X2+…+Xn) = nVar(X)
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15
Q

Discrete uniform distribution requirements and equation

A

This model is used when DUD X is defined over a set of n distinct values.
Each value is equally likely to occur, P(X=x)=1/n

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16
Q

Geometric distribution

A

Independent trials are carried out.
The outcome of each success is deemed a success or a failure.
The probability of success is the same for each trial.
The discrete random variable X is the number of trials needed to obtain the first successful outcome.
X~Geo(p)
P(X=r)=q^(r-1)*p

17
Q

Expectation, variance and others for geometric distribution

A

E(X)=1/p
Var(X)=q/p^2
P(X<=x)=1-q^x
P(X>x)=q^x

18
Q

Poisson distribution

A

Events occur singly and at random in a given time or space interval
λ is the mean number of occurrences in the interval and is known and finite.
P(X=x)=e^-λ*(λ^x)/x!

19
Q

Mode, expectation and variance of Poisson

A

If λ is an integer, mode = λ-1 and λ
If not, then mode = floor(λ)
E(X)=Var(X)=λ
Also, X~Po(m) and Y~Po(n) then X + Y~Po(m+n)

20
Q

When to use approximations

A

Binomial approx as poisson when n>50 and p<0.1, np<10
Binomial approx as normal when np>5 and nq>5
Poisson approx as normal when λ>15
Remember to do continuity corrections!

21
Q

Expectation and variance of a continuous variable

A

Integration must be used - formula for Var(X) is the same as normal.

22
Q

CDF for continuous

A

Integrate from a to t, where a is the lower limit.
When defining, remember to define outside the range as well.
To turn into a pdf, differentiate.

23
Q

Rectangular distribution equations

A

E(X)=1/2 (a+b)
Var(X) = 1/12 (b-a)^2
F(X)=(x-a)/(b-a)

24
Q

Normal distibution

A

Extends from -infinity to infinity
95% lies within 2 σ from μ
X~N(μ, σ^2)
F(X) for normal distributions is given by φ(z)

25
Q

How to standardise a normal variable

A

Z=(X-μ)/σ

26
Q

What is the formula for confidence interval?

A

k*σ/sqrt(n)

27
Q

What are the requirements to use PMCC?

A

The data sets are both random

28
Q

Define non-parametric test

A

A test where there is no underlying assumption that the data are from a normal distribution.

29
Q

What is the assumption for wilcoxon single and double sample test?

A

The distribution is symmetric about the median, and that the sample is random.

30
Q

When is a wilcoxon single rank approximated as normal?

A

When n is greater than 50

31
Q

What is the purpose of the Wilcoxon Rank-Sum test?

A

It checks whether two samples are drawn from the same distribution by seeing if their medians are the same.

32
Q

What are the assumptions of the wilcoxon rank-sum test?

A

X and Y are independent, and both distributions have the same shape