Continuous Random Variables and Their
Probability Distributions
Probability Distributions for Continuous Random Variables
We discussed discrete random variables and their distributions. We learned
that we can assign a positive probability to each possible value a discrete
random variable takes on. These values add up to 1. In the case of
continuous random variables, this is not possible anymore as the random
variable takes on infinitely many values with an interval or intervals and
the probabilities would not some to 1 anymore even if somehow we
manage to assign a probability to each value separately. We therefore can
no longer use the approach that we used for discrete random variables.
Examples of continuous random variables:
 Heights
 Weights,
 Length of life of a particular product
 Distance travelled on a trip
Probability Density Function
A probability density function (f(x)) is a model for the probability
distribution of a continuous random variable X.
The function f(x) is the probability density function for the continuous
random variable X if for all real numbers a and b, such that a < b, we have
P(a < X < b) is the area under the graph of f(x) between X= a and X = b.
The probability density function has the following properties:
 f(x)≥ 0.
 The area under the probability density function equals 1.
 P(X=a)=0. In other words, the probability at a specific point is equal
to zero.
 P(X ≥ a) = P(X > a) and P(X ≤ a) = P(X < a) as a result from the
previous point. (Remember that this is not true for discrete random
variables!!!!!).
Note: P(a < X < b) = P(a ≤ X< b) = P(a < X ≤ b) = P(a ≤ X ≤ b).
Exercise 1:
Consider the following probability density function for some random
variable X which can assume values in the interval [0,10].
f(x)
h
0
10
A. What is the height, h, of the function?
B. What is P(4 < X < 9)?
C. What is the probability that X is less than 1 or at least 4?
Uniform Distribution
The uniform random variable is used to model the behaviour of a
continuous random variable whose values are uniformly or evenly
distributed over a given interval.
A random variable X is said to have a uniform distribution on the real
interval [L,U] or X ~ U(L,U) if it has the PDF
1
f(x) = {𝑈−𝐿
0
𝐿+𝑈 2 (𝑈−𝐿)2
µ=
,σ =
2
12
𝐿≤𝑋≤𝑈
𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Exercise 2:
The amount of time it takes a clerk to process a certain type of application
follows a uniform distribution with the following density function
f(x) = {
0.5
0
4≤𝑋≤6
𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Find P(4.5<X<5.5), P(X=5), P(−2<X<6), P(X >5.5).
The Normal Probability Distribution
The normal distribution is perhaps the most famous continuous probability
distribution, as a large number of random variables naturally possess a
frequency distribution that is approximately mound-shape (normal
distribution).
Examples:
• Heights.
• Errors made in experimental measurements.
• The amount of rain received in a city.
In practice, we can have a normal random variable X that has any mean µ
and any variance σ2. We denote this by X ~ N(µ, σ2).
Then X has the following probability density function:
1
f ( x) =
e
 2
The Effect of the Mean
1  x− 
− 

2  
2
for −   x   .
Effect of the Variance
The Standard Normal Distribution
Calculating probability for a normally distributed random variable is
complicated as integrating the probability density function is quite
complex and time consuming. Tables are however available for a special
type of normal distribution called Standard Normal.
A standard normal random variable is typically denoted by the letter Z and
has the following probability density function:
f ( z) =
1
−z2 2
e
for −   z   .
2
Z has a mean of 0 and a variance of 1.
The graph of the pdf for the standard normal is shown below:
0.2
0.15
0.1
0.05
-3
-2
-1
1
2
3
Computing Probabilities with the Standard Normal Distribution
As with all continuous random variables, we can compute the probability
that Z takes on a value inside an interval like [a, b] – that is, P(a < Z < b)
– by simply computing the area under the probability density function
between a and b.
However, to do this with the probability density function f(z) would require
integration.
Instead, we will use a table of already calculated probabilities of the form
P(Z ≤ z). The table is available on brightspace.
Exercise 3:
Use the standard normal table to compute the following probabilities.
A. P(Z ≤1.12) B. P(Z < 1.12) C. P(Z > 0.97) D. P(Z > –2.43)
What if the probability required is of the form P(a ≤ Z ≤ b)?
We can
rewrite the probability as
P(a ≤ Z ≤ b) = P(Z ≤ b) − P(Z ≤ a)
Exercise 4:
Use the standard normal table to compute the following probabilities:
P(−1.16 ≤ Z ≤ 2.97), P(−1 ≤ Z ≤ 1)
Calculating Probabilities for a General Normal Random
Variable
Most of the normal random variables that you are dealing with are not
standard normal. So to calculate the probability for a general normal
random variable you will need to standardize it ( i.e., turn it into a standard
normal). This can be done as follows:
2
If X~N(µ,σ ) then Z =
𝑋−𝜇
𝜎
~ N(0,1)
Exercise 5:
Assume a driver traveling at a speed of 50km/hr will react instantly once
they see an obstacle in front of their car. The distance it takes them to stop
has a normal distribution with a mean of 3m and a variance of 0.25m2.
If a person walks onto the road and the car is 4.2m away, what is the
probability they will not get hit?
Working Backwards
Given a value of a standard normal random variable z, we now know how
to find the area to the left and right of it.
What if, instead, we are given something of the form P(Z < z) = 0.7939.
Could we say what z is equal to? In other words, could we find a standard
normal value z that has an area of 0.7939 to the left of it? We have to use
the Z table to work backwards!
Exercise 6:
Solve for z in each case below.
P(Z < z) = 0.7939, P(Z > z) = 0.8186, P(– z < Z < z) = 0.7540
Exercise 7:
The air pressure in a randomly selected tire put on a certain car is normally
distributed with mean 31lb/in2 and standard deviation 0.2lb/in2.
th
A. What is the 99 percentile of the tire pressure distribution?
B. Give an interval such that 95% of all air pressure readings will be
within that interval.