S2 - Chapters 1-5

Question 1

Binomial Distribution Function

Answer 1

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

Question 2

Binomial Distribution Conditions

Answer 2

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”

Question 3

Binomial P(X=x) =

Answer 3

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

Question 4

Factorial Function n!

Answer 4

gives us the number of ways of arranging distinguishable objects

Question 5

Choose Function

Answer 5

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

Question 6

Cumulative Distribution Function notation

Answer 6

F(x) = P(X

Question 7

Cumulative Distribution Function meaning

Answer 7

the probability up to a particular value

Question 8

Binomial Distribution Mean

Answer 8

E(X) = np

Question 9

Binomial Distribution Variance

Answer 9

o^2 = np(1-p)

Question 10

Binomial Distribution Description

Answer 10

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

Question 11

Poisson Distribution Description

Answer 11

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

Question 12

Poisson P(X=x) =

Answer 12

e^ - lambda x lambda^x / x!

Question 13

Poisson Distribution Conditions

Answer 13

Events must occur:

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

Question 14

Poisson Distribution Mean

Answer 14

E(X) = lambda

Question 15

Poisson Distribution Variance

Answer 15

o^2 = lambda

Question 16

Approximating a Binomial using a Poisson

Answer 16

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

Question 17

Poisson Distribution Function

Answer 17

X ~ Po (lambda)

Question 18

Types of discrete random variables

Answer 18

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

Question 19

Types of continuous random variables

Answer 19

normal distributions

Question 20

Probability Density Function (p.d.f)

Answer 20

f(x)

Question 21

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

Answer 21

• f(x) >- 0

- P(a

Question 22

Cumulative Distribution Function

Answer 22

F(x) = P(X

Question 23

F(x) to f(x)

Answer 23

f(x) = differentiate F(x)

Question 24

f(x) Mean (if continuous)

Answer 24

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

Question 25

f(x) Variance (if continuous)

Answer 25

o^2 = E(X^2) - E(X)^2
o^2 = integrate f(x) times x^2 - mean^2

Question 26

Mean of F(x)

Answer 26

F(m) = 0.5

- find the range where the median x value is in and then set the function equal to 0.5

Question 27

Lower Quartile of F(x)

Answer 27

F(Q1) = 0.25

Question 28

Upper Quartile of F(x)

Answer 28

F(Q3) = 0.75

Question 29

Positive Skew

Answer 29

mode < median < mean

Question 30

Negative Skew

Answer 30

mode > median > mean

Question 31

Continuous Uniform Distribution f(x)

Answer 31

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)

Question 32

Continuous Uniform Distribution Mean

Answer 32

E(x) = a + b / 2

Question 33

Continuous Uniform Distribution Variance

Answer 33

(b-a)^2 / 12

Question 34

E(X^2) =

Answer 34

Var(X) + E(X)^2

- because Var(X) = E(X^2) - E(X)^2

Question 35

Continuous Uniform Distribution F(x)

Answer 35

integrate 1 / (b-a) which turns into x-a / b-a

Question 36

Continuity Corrections

Answer 36

• 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

Question 37

Approximating a Binomial using a Normal

Answer 37

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

Question 38

Approximating a Poisson using a Normal

Answer 38

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