Binomial and Geometric Distributions

Question 1

Conditions of Binomial Distribution

conditions
parameter notation
def

Answer 1

Binomial distribution is a common type of probability distribution which obeys conditions (4)

1. Only 2 possible outcomes (discrete) usually called successes (p) or failures (q), [successes (p), outcome we are interested]
2. There are a fixed number of trials (n)
3. Each trial must be independent of the other trials
4. The probability of success (p) is fixed at each trial

n and p are called parameters of the distribution:
X ~ B (n, p)

Definition: A binomial distribution is used to model the number of success in a fixed number of independent trials

Question 2

Using the binomial distribution as a model

Answer 2

Before using binomial distribution as a model for a situation you need to convince yourself that all the conditions are satisfied.

Question 3

Binomial distribution formula

Answer 3

P(X = x) = (n x*)pˣ(1-p)ⁿ⁻ˣ
or
P(X = x) = (n x*)pˣqⁿ⁻ˣ

• x placed below n
• same as nCr on calculator

You need to know n and p, q isn’t essential as q is equal to 1 - p

Question 4

Alternative methods for Binomial Distribution

expected value
variance
standard deviation

Answer 4

E(X) = np

Var(X) = npq

Sd(X) = √Var(X) = √npq

Question 5

Mode of Binomial Distribution

mode
in probability distribution diagram

Answer 5

The mode is the value of X that is most likely to occur, the X value with the highest probability

In a vertical line diagram (probability distribution diagram), the mode is the value represented by the highest line

Question 6

Conditions of Geometric Distribution

conditions
parameter notation
def

Answer 6

Geometric distribution is a common type of probability distribution which obeys conditions (4)

1. Only 2 possible outcomes (discrete) usually called successes (p) or failures (q), [successes (p), outcome we are interested]
2. The repeated trials can be infinite
3. Each trial must be independent of the other trials
4. The probability of success (p) is fixed at each trial

n and p are called parameters of the distribution:
X ~ Geo (p)

Definition: A geometric distribution is used to model the number of trials up to and including the first success in an infinite number of independent trials

Question 7

Geometric distribution Formula

general formula
simplified formula
expected value formula
variance formula

Answer 7

P(X = x) = p(1 - p)ˣ⁻¹
or 
P(X = x) = pqˣ⁻¹

E(X) = 1/p
Var(X) = q/p²

Question 8

Binomial Distribution vs Geometric Distribution

comparison
binomial
geometric

Answer 8

For binomial distribution the number of trials is fixed from the start and the number of successes are counted, in geometric distribution trials are repeated as many times as necessary until the first success occurs

For X ~ B (n, p) there are ⁿCᵣ ways to obtain x successes

For X ~ Geo(p) there is only one way to obtain the first success on the xᵗʰ trial, that is when there are x-1 failures followed by a success

Question 9

Geometric Distribution with Inequalities

fewer than formula
after formula

Answer 9

P(X ≤ x)
= 1 - (1 - p)ˣ
= 1 - qˣ

P(X > x)
= (1 - p)ˣ
= qˣ

Question 10

Mode of Geometric distribution

common features

Answer 10

All geometric distribution have two common features:

• P(X = 1) has the greatest probability in all geometric distributions meaning X ~ Geo(p) is X = 1
• P(X = x) decreases as x increases