Binomial and Normal Distribution properties

Question 1

Binomial random variable

Answer 1

A random process with just two possible outcomes. One outcome is arbitrarily labelled success and the other a failure.

Examples:

1. Coin toss
2. Pass-or-fail test

Question 2

Bernoulli trial properties

Answer 2

1. The result of each trial is either a success or a failure.
2. The probability p of a success is the same in every trial.
3. The trials are independent; ie, the outcome of each trial has no influence on later outcomes.

Question 3

Bernoulli random variable

Answer 3

X is the number of success in n Bernoulli trails with probability p of success.

Question 4

Mean and variance of a binomial distribution

Answer 4

• µ= np*
• σ² = np(1-p)*

Question 5

Parameters of a binomial distribution

Answer 5

n and p

where n is the number of trials and p is the probability of success.

Question 6

Relationship between binomial distribution and the standard normal distribution

Answer 6

The binomial distribution is closely approximatedy by the standard normal distribution when p=0.5 and n is very large.

Question 7

Steps for transforming a binomial distribution into a standard normal distribution

Answer 7

1. Set p to be 0.5
2. Make n tend towards infinity.
3. Set the mean to be 0 by a linear transformation (ie making it symmetrical about the y axis.
4. Set the highest y value such that the area under the curve is 0.