Probability Distributions

Probability Distributions

A. Discrete Random Variables

1. A random variable that has finite and countable values is known as a discrete random variable.

2. For example, two coins are tossed simultaneously and the number of tails obtained is studied. If X represents the number of tails obtained, then X can take the values 0, 1 and 2 based on the following table.

Outcomes	TT	TH	HT	HH
X	2	1	1	0

Probability of an Event that follows a Binomial Distribution

1. A trial with only two outcomes i.e. 'success' or 'failure' is known as a Bernoulli trial.

2. If a Bernoulli trial is repeated many times, the experiment is known as the binomial experiment.

3. Let X be a discrete random variable that represents the number of success of a binomial experiment. X follows a binomial distribution (with the number of experiments = n and the probability of success = p) and it is denoted by

Example

Mean, Variance and Standard Deviation of a Binomial Distribution

If X is a binomial discrete random variable such that X~B (n,p), then

• Mean of X = np
• Variance of X = npq
• Standard deviation of X = √ npq

B. Normal Distribution

1. Continuous random variable is a variable that can take any infinite value in a certain range.

2. For example, in a Form 4 class, the mass of the heaviest student is 70 kg and the mass is lightest student is 50 kg. If X represents the masses of any student in that class, then X can take any value from 50 kg to 70 kg such as 60 kg, thus

X = { x: 50 kg ≤ x ≤ 70 kg, x is the mass of students }

3. If a normal random variable, X, has a mean, µ = 0 and a standard deviation σ = 1, then X follows a standard normal distribution, i.e. X~N (0,1).

 4. The variable of a normal distribution can be converted to the variable of the standard normal distribution using the following formula.

                                                                                                                                                
z-Score of a Normal Distribution


  

To Read the Standard Normal Distribution Tables

Probability of an Event that follows a Normal Distribution

To Find Mean and Standard Deviation of a Normal Distribution

Further Examples on Normal Distrbution