S2 - Chapters 1-5 Flashcards

1
Q

Binomial Distribution Function

A

X ~ B(n, p)
n = number of successes
p = probability of success

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2
Q

Binomial Distribution Conditions

A

1) FIXED number of trials
2) INDEPENDENT trials
3) CONSTANT probability of success (in each trial)
4) TWO outcomes in each trail - “success” and “failure”

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3
Q

Binomial P(X=x) =

A

(n choose X) x P^X x (1-p)^n-X

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4
Q

Factorial Function n!

A

gives us the number of ways of arranging distinguishable objects

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5
Q

Choose Function

A

(n choose r) = n! / (r! (n-r)! )

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6
Q

Cumulative Distribution Function notation

A

F(x) = P(X

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7
Q

Cumulative Distribution Function meaning

A

the probability up to a particular value

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8
Q

Binomial Distribution Mean

A

E(X) = np

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9
Q

Binomial Distribution Variance

A

o^2 = np(1-p)

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10
Q

Binomial Distribution Description

A

The number of “successes” out of n trials, each with a probability of p of success

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11
Q

Poisson Distribution Description

A

a distribution over the number of events which occur within a period of time, give an average rate lambda

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12
Q

Poisson P(X=x) =

A

e^ - lambda x lambda^x / x!

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13
Q

Poisson Distribution Conditions

A

Events must occur:

1) SINGLY in time
2) INDEPENDENTLY of each other
3) at a CONSTANT AVERAGE RATE

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14
Q

Poisson Distribution Mean

A

E(X) = lambda

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15
Q

Poisson Distribution Variance

A

o^2 = lambda

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16
Q

Approximating a Binomial using a Poisson

A

If X ~ B(n,p) and:
- n is large
- p is small
The X can be approximated by X ~ Po(np)

17
Q

Poisson Distribution Function

A

X ~ Po (lambda)

18
Q

Types of discrete random variables

A

1) Binomial distribution
2) Poisson distribution
3) Discrete uniform distribution

19
Q

Types of continuous random variables

A

normal distributions

20
Q

Probability Density Function (p.d.f)

21
Q

If X is a continuous random variable with p.d.f. f(x) then:

A
  • f(x) >- 0

- P(a

22
Q

Cumulative Distribution Function

A

F(x) = P(X

23
Q

F(x) to f(x)

A

f(x) = differentiate F(x)

24
Q

f(x) Mean (if continuous)

A

E(x) = integrate f(x) times x

25
f(x) Variance (if continuous)
``` o^2 = E(X^2) - E(X)^2 o^2 = integrate f(x) times x^2 - mean^2 ```
26
Mean of F(x)
F(m) = 0.5 | - find the range where the median x value is in and then set the function equal to 0.5
27
Lower Quartile of F(x)
F(Q1) = 0.25
28
Upper Quartile of F(x)
F(Q3) = 0.75
29
Positive Skew
mode < median < mean
30
Negative Skew
mode > median > mean
31
Continuous Uniform Distribution f(x)
basically a straight horizontal line running from the x value range and you find probability by finding the area under the line. - the y line is 1 / (b-a)
32
Continuous Uniform Distribution Mean
E(x) = a + b / 2
33
Continuous Uniform Distribution Variance
(b-a)^2 / 12
34
E(X^2) =
Var(X) + E(X)^2 | - because Var(X) = E(X^2) - E(X)^2
35
Continuous Uniform Distribution F(x)
integrate 1 / (b-a) which turns into x-a / b-a
36
Continuity Corrections
- approximate a discrete distribution from a continuous one - round up or down by 0.5 - etc. we can't find exact probabilities like P(Y=6) when Y is continuous because the probability is effectively 0 - P(5.5
37
Approximating a Binomial using a Normal
To approximate, we just copy over the mean and variance of the Binomial to the Normal If n is large and p is close to 0.5 X ~ B(n,p) --> Y ~ N(np = mean, np(1-p) = variance) then you use normal distribution to find the probability like using the equation z = x - m / o
38
Approximating a Poisson using a Normal
To approximate, we use the same mean and variance for the Normal as the original poisson If lambda is large X ~ Po(lambda) --> Y ~ N(lambda = mean, lambda = variance) then you use normal distribution to find the probability like using the equation z = x - m / o