Discrete probability distributions

Question 1

Conditions for binomial distribution

Answer 1

Fixed number of trials
fixed chance of success
two outcomes only
independent success rate of trials

Question 2

conditions of poisson distribution

Answer 2

mean rate of occurrence in a given time period
random event
events are independent of each other

Question 3

Binomial notation

Answer 3

X ~ B(n, p)
where n is the number of trials and p is the chance of success

Question 4

Binomial formula

Answer 4

P (X = r) = nCr x p^r x q^n-r

Question 5

E(X) for binomial distribution

Answer 5

E(X) = np

Question 6

Var(X) for binomial distribution

Answer 6

Var(X) = npq
or np - np^2

Question 7

poisson formula

Answer 7

P(X = r) = e^-λ x λ^r/r!
where λ = the mean rate of occurrence

Question 8

poisson notation

Answer 8

X ~ Po(λ)
λ i.e mean rate of occurrence

Question 9

poisson skew

Answer 9

positively skewed
has less of an effect as λ increases

Question 10

what is a unique feature of poisson distribution

Answer 10

E(X) = Var(X) = λ
good for questions which ask what type of distribution is appropriate

Question 11

If X ~ Po(x)
and Y ~ Po(y)
Then what is X + Y

Answer 11

X + Y ~ Po(x + y)

Question 12

Geometric distribution criteria

Answer 12

1 outcome, fixed and independent chance

Question 13

expectation and variance of x

Answer 13

E = 1/p
Var = (1-p)/ p^2

Question 14

Geometric distribution p values

Answer 14

P(X = x) - (1-p)^x-1 x p
P(X <= x) = 1-(1-p)^x
P(X > x) = (1-p)^x

Question 15

Uniform distribution criteria

Answer 15

All outcomes have an even chance
only works if evenly distributed from a to b

Question 16

Expectation and variance of x = 1….n

Answer 16

E = n+1)/2
Var = 1/12(n+1)(n-1)

Question 17

Working out E and Var for x = a….b

Answer 17

E = a+b)/2
Var (and for E) - scale up the original formula 1…n then multiply / add var and e accordingly