MTH5P2A: Continuous Random Variables

This unit explains Continuous Random Variables

Notation.
The indicator function of a set S is a real-valued function defined by :

Suppose that is a real-valued function whose domain is an arbitrary set D. The support of f, written supp(f), is the set of points in D where f is nonzero

supp(f) = {x 2 D| f(x) 6= 0}.

Probability Density Function and Cumulative Distribution Function
Definition 1.1 (Probability density function). A rrv is said to be (absolutely) continuous if there exists a real-valued function fX such that, for any subset

Then fX is called the probability density function (pdf) of the random variable X.
In particular, for any real numbers a and b, with a < b, letting B = [a, b], we obtain from Equation (1) that :

If X is a continuous rrv, then

In other words, the probability that a continuous random variable takes on any fixed value is zero.

For any real numbers a and b, with a < b

The above equation states that including or not the bounds of an interval does not modify the probability of a continuous rrv.

Let us first prove Equation (3) :

To prove Equation (4), we simply notice that

Theorem 1.1. A probability density function completely determines the distribution of a continuous real-valued random variable.
Remark : This theorem means that two continuous real-valued random variables X and Y that have exactly the same probability density functions follow the same distribution. We say that they are identically distributed.
Definition 1.2 (Cumulative distribution function). Let ( ,A, P) be a probability space. The (cumulative) distribution function (cdf) of a real-valued random variable X is the function FX given by

Let FX be the cdf of a random variable X. Following are some properties of FX:

(Cumulative distribution function of a continuous rrv). Let X be a continuous rrv with pdf fX. Then the cumulative distribution function FX of X is given by :

Proof. We have the following :

A cumulative distribution function completely determines the distribution of a continuous real-valued random variable.

Let X be a continuous rrv with pdf fX and cumulative distribution function FX. Then

For any real numbers a and b with a < b,

Proof. This is a direct application of the Fundamental Theorem of Calculus.

Let X be a continuous rrv on probability space with pdf fX. Then, we have :
fX is nonnegative on R :

fX is integrable on R and

Proof of (8) : Property 1.2 states that the cumulative distribution function FX is increasing on R. Therefore According to Lemma 1.3, is continuous at x. This completes the

Proof of (9) :

In other words, the event that X takes on some value is the sure event 

Definition 1.3. A real-valued function f is said to be a valid pdf if the following holds :
• f is nonnegative on R :

• f is integrable on R and

This means that if f is a valid pdf, then there exists some continuous rrv X that has f as its pdf.

Example. A continuous rrv X is said to follow a uniform distribution on [0, 1/2] if its pdf is :

Questions.
(1) Determine c such that fX satisfies the properties of a pdf.
(2) Give the cdf of X.

Proof. (1) Since is a pdf, should be nonnegative for all This is the case for where fX(x) equals zero. On the interval [0, 1/2], fX(x) = c. This implies that c should be nonnegative as well.

Let us now focus on the second condition,

And we check that indeed c = 2 is nonnegative.
(2) The cumulative distribution function FX of X is piecewise like its pdf :

Example. Let X be the duration of a telephone call in minutes and suppose X has pdf :

Questions.
(1) Which value(s) of c make(s) fX a valid pdf?
Answer. c = 1/10.
(2) Find the probability that the call lasts less than 5 minutes.

Expectation and Variance

Definition 2.1 (Expected R Value). Let X be a continuous rrv with pdf fX. If  dx is absolutely convergent, i.e. then, the mathematical expectation (or expected value or mean) of X exists, is denoted by and is defined as follows :

Definition 2.2 (Expected Value of a Function of a Random Variable). Let X be a continuous rrv with pdf  be a piecewise continuous function.
If random variable g(X) is integrable.Then, the mathematical expectation of g(X) exists, is denoted by and is defined as follows :

Example. Compute the expectation of a continuous rrv X following a uniform distribution on [0, 1/2]. As seen earlier, its pdf is given by:

Let X be a continuous rrv with pdf fX.

We then say that X is integrable.

If g :  is a nonnegative piecewise continuous function and g(X) is integrable. Then, we have :

If g1 : are piecewise continuous functions and g1(X) and g2(X) are integrable such that

Then, we have :

Proof. • Proof of Equation (14) : Here we consider the function of the random variable X defined by : g(X) = c. We then get :

Proof of Equation (15) : This comes from the nonnegativity of the integral for nonnegative functions.
Proof of Equation (16) : This is a direct application of Equation (15) applied to function g2 − g1.

Property 2.2 (Linearity of Expectation). Let X be a continuous rrv with pdf are piecewise continuous functions and g1(X) and g2(X) are integrable. Then, we have :

Note that Equation (17) can be extended to an arbitrary number of piecewise continuous functions.
Proof. 

Definition 2.3 (Variance–Standard Deviation). Let X be a real-valued random variable. When the variance of X is defined as follows :

Property 2.3. The variance of a real-valued random variable X satisfies the following properties :

Proof. This property is true for any kind of random variables (discrete or continuous). See proof of Property 4.1 given in the lecture notes of the chapter about discrete rrvs.

Theorem 2.1 (König-Huygens formula). Let X be a real-valued random variable. When exists, the variance of X is also given by :

Proof. This property is true for any kind of random variables (discrete or continuous). See proof of Theorem 4.1 given in the lecture notes of the chapter about discrete rrvs.

In the case of continuous random variables, Equation (20) becomes :

Example. Compute the variance of a continuous rrv X following a uniform distribution on [0, 1/2]. As seen earlier, its pdf is given by:

Proof. We computed previously the expectation of X that is E[X] = 1/4. Computing the variance of X thus boils down to calculating

Common Continuous Distributions

Uniform Distribution
Definition 3.1. A continuous rrv is said to follow a uniform distribution U(a, b) on a segment [a, b], with a < b, if its pdf is

Proof. Let us prove that the pdf of a uniform distribution is actually a valid pdf :

Property 3.1 (Mean and Variance for a Uniform Distribution). If X follows a uniform distribution U(a, b), then
its expected value is given by :

its variance is given by :

Proof. • Expectation :

Then using the shortcut formula (20), we find :

Gaussian Distribution
Example. A restaurant wants to advertise a new burger they call The Quarterkilogram.
Of course, none of their burgers will exactly weigh exactly 250 grams. However, you may expect that most of the burgers’ weights will fall in some small interval centered around 250 grams.

Definition 3.2. A continuous random variable is said to follow a normal (or Gaussian) distribution then

its expected value is given by :

its variance is given by :

The normal distribution is one of the most (even perhaps the most) important distributions in Probability and Statistics. It allows to model many natural, physical and social phenomenons. We will see later in this course how
all the distributions are somehow related to the normal distribution.

Property 3.3. If X follows a normal distribution  Then its pdf fX has the following properties:

Definition 3.3 (Standard Normal Distribution). We say that a continuous rrv X follows a standard normal distribution if X follows a normal distribution  N(0, 1) with mean 0 and variance 1. We also say that X is a standard normal random variable. In particular, the cdf of a standard normal random variable is denoted that is :

Property 3.4. The cdf of a standard normal random variable satisfies the following property:

Property 3.5. Let a  If X follows a normal distribution then random variable aX + b follows a normal distribution N(aμ +

Proof. Let us assume that a > 0. Consider random variable Y = aX+b. We will prove that FY , the cdf of Y is the cdf of a normal distribution we have that :

Now, consider the change of variable : t = ax + b ) dt = a dx. Hence, we have

We now recognize the cdf of a normal distribution which completes the proof.
The proof in the case where a is negative is left as an exercise.

Corollary 3.1. If X follows a normal distribution then random variable Z defined by:

is a standard normal random variable.

Finding Normal probabilities. If The above property leads us to the following strategy for finding probabilities

Transform X, a, and b, by:

Use the standard normal N(0, 1) Table to find the desired probability.

Example. Let X be the weight of a so-called ‘quarter-kilogram’ burger. Assume X follows a normal distribution with mean 250 grams and standard deviation 15 grams.
(a) What is the probability that a randomly selected burger has a weight below 240 grams?
(b) What is the probability that a randomly selected burger has a weight above 270 grams?
(c) What is the probability that a randomly selected burger has a weight between 230 and 265 grams?

Quantiles
Previously, we learned how to use the standard normal curve N(0, 1) to find probabilities concerning a normal random variable X. Now, what would we do if we wanted to find some range of values for X in order to reach some probability?
Definition 3.4 (Quantile). Let X be a rrv with cumulative distribution function
A quantile of order for the distribution of X, denoted ,is defined as follows :

The quantile of order 1/2 is called the median of the distribution.

If FX is a bijective function, then 

Property 3.6. For the quantile function of a standard normal random variable is given by :

Finding the Quantiles of a Normal Distribution. In order to find the value of a normal random variable

Find in the Table the value associated with the desired probability.

Example. Suppose X, the grade on a midterm exam, is normally distributed with mean 70 and standard deviation 10.
(a) The instructor wants to give 15% of the class an A. What cutoff should the instructor use to determine who gets an A?
(b) The instructor now wants to give the next 10% of the class an A-. For which range of grades should the instructor assign an A-?

Exponential Distribution
Motivating example. Let Y be the discrete rrv equal to the number of people joining the line to visit the Eiffel Tower in an interval of one hour. If x, the mean number of people arriving in an interval of one hour, is 800, we are interested in the continuous random variable X, the waiting time until the first visitor arrives.
Definition 3.5. A continuous random variable is said to follow an exponential distribution is given by:

Property 3.7 (Mean and Variance for an Exponential Distribution). If X follows an exponential distribution then
its expected value is given by :

its variance is given by :

Example. Suppose that people join the line to visit the Eiffel Tower according to an approximate Poisson process at a mean rate of 800 visitors per hour. What is the probability that nobody joins the line in the next 30 seconds?.

ASSIGNMENT : Continuous Random Variables Assignment MARKS : 50  DURATION : 1 week, 3 days

 

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