Statistics for Data Science -1
Statistics for Data Science -1
Continuous Random Variables
Usha Mohan
Indian Institute of Technology Madras
1/ 15
Statistics for Data Science -1
Learning objectives
2/ 15
Statistics for Data Science -1
Learning objectives
1. Define what is a continuous random variable.
2/ 15
Statistics for Data Science -1
Learning objectives
1. Define what is a continuous random variable.
2. Probability distribution function and examples
2/ 15
Statistics for Data Science -1
Learning objectives
1. Define what is a continuous random variable.
2. Probability distribution function and examples
3. Cumulative distribution function, graphs, and examples.
2/ 15
Statistics for Data Science -1
Learning objectives
1. Define what is a continuous random variable.
2. Probability distribution function and examples
3. Cumulative distribution function, graphs, and examples.
4. Expectation and variance of random variables.
2/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
3/ 15
Statistics for Data Science -1
Discrete and Continuous random variables
4/ 15
Statistics for Data Science -1
Discrete and Continuous random variables
Definition
A random variable that can take on at most a countable number of
possible values is said to be a discrete random variable.
4/ 15
Statistics for Data Science -1
Discrete and Continuous random variables
Definition
A random variable that can take on at most a countable number of
possible values is said to be a discrete random variable.
Definition
When outcomes for random event are numerical, but cannot be
counted and are infinitely divisible, we have continuous random
variables.
4/ 15
Statistics for Data Science -1
Discrete and continuous random variable
5/ 15
Statistics for Data Science -1
Discrete and continuous random variable
I A discrete random variable is one that has possible values that
are discrete points along the real number line.
5/ 15
Statistics for Data Science -1
Discrete and continuous random variable
I A discrete random variable is one that has possible values that
are discrete points along the real number line.
I A continuous random variable is one that has possible values
that form an interval along the real number line. In other
words, a continuous random variable can assume any value
over an interval or intervals.
5/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Probability density function (pdf)
I Every continuous random variable X has a curve associated
with it.
6/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Probability density function (pdf)
I Every continuous random variable X has a curve associated
with it.
I The probability distribution curve of a continuous random
variable is also called its probability density function. It is
denoted by f (x)
6/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Area under a pdf
I Consider any two points a and b, where a is less than b.
I The probability that X assumes a value that lies between a
and b is equal to the area under the curve between a and b.
That is,
P(X ∈ [a, b]) = P(a ≤ X ≤ b) is area under curve between a and b
Z b
P(a ≤ X ≤ b) = f (x)dx
a
7/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Properties of pdf
8/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Properties of pdf
1. The area under the probability distribution curve of a
continuous random variable between any two points is
between 0 and 1.
8/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Properties of pdf
9/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Properties of pdf
2. Total area under the probability distribution curve of a
continuous random variable is always 1.
9/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Properties of pdf
I The area under the graph of the probability density function
between points a and b is the same regardless of whether the
endpoints a and b are themselves included:
P(a ≤ X ≤ b) = P(a < X < b)
10/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Properties of pdf
I The area under the graph of the probability density function
between points a and b is the same regardless of whether the
endpoints a and b are themselves included:
P(a ≤ X ≤ b) = P(a < X < b)
I The probability density curve of a random variable X is a
curve that never goes below the x− axis
10/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
2. Less than 40
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
2. Less than 40 =0.56
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
2. Less than 40 =0.56
3. More than 50
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
2. Less than 40 =0.56
3. More than 50 =0.33
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
2. Less than 40 =0.56
3. More than 50 =0.33
4. Between 40 and 70 minutes to
complete a repair?
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Example
Figure below is a probability density function for the random
variable that represents the time (in minutes) it takes a repairer to
service a television. The numbers in the regions represent the areas
of those regions.
What is the probability that the repairer
takes
1. Less than 20 =0.29
2. Less than 40 =0.56
3. More than 50 =0.33
4. Between 40 and 70 minutes to
complete a repair? =0.27
11/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Cumulative distribution function
For a continuous random variable X
Z a
F (a) = P(X ≤ a) = f (x)dx
−∞
Since the probability that a continuous random variable X assumes
a single value is always zero, we have
Z a
P(X < a) = P(X ≤ a) = f (x)dx
−∞
12/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Expectation and Variance
R
I Expected value: E (X ) = x f (x)dx.
I Variance: Var (X ) = (x − E (X ))2 f (x)dx
R
13/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Section summary
I Probability density function and its properties.
I cdf, expectation, and variance of continuous random variables.
14/ 15
Statistics for Data Science -1
Probability density function, graph, and examples
Probability density function
Introduction
I A random variable is said to be uniformly distributed over the
interval [0, 1] if its probability density function is given by
1 0≤x ≤1
f (x) =
0 otherwise
15/ 15
