This document discusses discrete and continuous random variables. A discrete random variable has a finite or countable number of possible values and can be summarized with a probability distribution listing each value and probability. The mean and variance of a discrete variable are calculated by weighting each possible value by its probability. A continuous random variable has an uncountable number of possible values, so its probability distribution is represented with a density curve rather than a histogram, and calculating probabilities requires integration of the density curve.

1. By Ishara .S.Saranapala

2. Discrete Random Variables
Continuous Random Variables

3.  A random variable is a numerical outcome
of a random process or random event
 Example: three tosses of a coin
• S = {HHH,THH,HTH,HHT,HTT,THT,TTH,TTT}
• Possible values for X = {0,1, 2, 3}
• What is the importance of random
variables?
• We use them as a model for our observed data

4.  A ‘Discrete random variable’ has a finite or
countable number of distinct values and
‘Discrete random variables’ can be
summarized by listing all values along with
the probabilities
• Called a Probability Distribution

5.  Random variable X = the sum of two dice
• X takes on values from 2 to 12
 If discrete r.v. takes on many values, it is
better to use a Probability Histogram

6. Ex: Probability histogram of sum of two dice:
 Using the disjoint addition rule, probabilities
for discrete random variables are calculated
by adding up the “bars” of this histogram:
P(sum > 10) = P(sum = 11) + P(sum = 12) = 3/36

7.  Mean is the sum of all possible values, with
each value weighted by its probability:
μ = Σ xi*P(xi) = x1*P(x1) + … + x12*P(x12)

8.  Variance is the sum of the squared deviations
away from the mean of all possible values,
weighted by the values probability:
μ = Σ(xi-μ)2*P(xi) = (x1-μ)2*P(x1) + … +(x12-
μ)2*P(x12)
 Example: X = sum of two dice

9.  Continuous random variables have un-
countable number of values
 Can’t list the entire probability distribution,
so we use a Density curve instead of a
histogram
 Eg. Normal density curve:

10.  Discrete case: adding up bars from probability
histogram
 Continuous case: we have to use Integration to
calculate the area under the density curve: