Embedded Files
Chapter S3: Discrete Random Variable (DRV) - Session 2
INSTRUCTIONS:
1. This topic is designed as a self-contained learning unit.
2. This is to be covered over TWO Blended Learning sessions.
3. You should have the notes for Chapter S3 with you.
You are to complete Session 2 by 04 March 2025.
You should set aside 1 hour for this BL session.
Pre-Learning:
1. Recap Session 1 learning activity (with the Appendix).
You would need to recall definitions of expectation (Section 3.2) and how to compute expectation and square of standard deviation (Section B of appendix).
This would lead to the concept of variance (Section 3.3). As you view to the lectures for section 3.3, try make connections to the concept of standard deviation (Sec 3/4).
NOTE THAT the variance concept (square of standard deviation) is a useful term to refer to in addition to standard deviation.
Self-check (from BL Session 1):
For the appendix learning activity Section (A) Computing Mean and Square of Standard Deviation from Experimental Data, your computations should show:
(a) For mean, values to be in the range of 3.3 to 3.7.
(b) For square of SD, values should be close to 2.9.
If you did not get these values, you are expected to do your calculations again. Please note the experiment is for 60,000 rolls of a die.
(T2W4) Appendix BL Activity -(Answers).pdfLearning:
1. Watch and fill in your lecture notes for Ch S3 Section 3.3.
Note: Reference the Appendix activity you did in Session 1, the way we compute Variance of X was using the definition and the calculations can be rather tedious and cumbersome. With Section 3.3 Variance, we now have a simpler way to calculate variance.
Thus, you need to learn to compute variance in the method detailed in Example 10 using
Var (X) = E(X^2) – (E(X))^2 .
VISUAL REPRESENTATION of PROPERTIES OF VARIANCE
The following section serve as visual representation to understand the properties of variance and is a good explanation as to why
(i) Var(a) = 0 ,
(ii) Var( X + a) = Var(X) and
(iii) Var(aX)= a^2 Var(X)) , where a is a constant.
View the video. (7:00min onwards explains the Var(3X) scenario).
For a more detailed explanation with static screenshots, please see below.
(i) Var(a) = 0 [Variance of a constant = 0]. See the following Diagram (i)
Note and see that the data points are all 10 each, the mean is 10.
Think about the spread? Do you see any spread of data?
Now I think that since there is no spread of the data, standard deviation = 0, and hence variance = 0.
(ii) Var(X + a) = Var(X) ,
Var (X –a) = Var(X)
Given that the random variable X takes on values 1,3,4,8, as represented by the red dots.
The mean is 4, with a certain spread.
See Diagram (ii)(a) for how the data points spread.
Now, let a = 3, and consider the random variable X+3.
Var(X + 3) = Var(X)
After considering the X+3 values (i.e., takes on values 4, 6, 7, 11), note that each of the values move up by 3 respectively, as in the diagram (ii)(b).
The new mean is 7 (represented by the red line), and the spread remains the same as before.
This gives a visual representation that adding a constant to a variable does not change the spread of the data , hence its variance does not change.
Similarly Var( X - 3) = Var (X).
(iii) Var(3X) = 3^2 Var(X) = 9 Var(X)
Now we consider a new variable 3X based on the previous scenario of X taking on 1,3,4,8.
See Diagram (iii).
Observe that the original data points (X=1,3,4, and 8) represented by red points are now (3X = 3,9,12, and 24) represented by blue points.
The mean changes from 4 (for X) to 12 (for 3X) (red and blue line respectively), which means a multiple of 3 from original X as 3 x 4 = 12.
ACKNOWLEDEGMENTS
Note that the standard deviation from the mean has increased substantially by a factor of 3 too. Note the distance of the blue dots (random variable 3X) from blue line (as seen from the grid lines) is THREE times the distance of the red dots (r.v. X) from the red line.
For example, red point (8) is 4 units (or 2 grid lines) from the red-line and corresponding blue point (24) is 12 units (or 6 grid lines) from the blue-line.
The numerical values also showed that SD of 3X is 3 times that of X. Since variance is square of deviation, the result follows that Var(3X) = 3^2 Var(X) = 9 Var(X).
LEARNING:
Lecture Notes Sections 3.4 and 3.5
Watch this and fill in your lecture notes: Ch S3 Section 3.4.
Note: This section is very important and you would need to manipulate the expectation and variance of various independent random variables if they are combined.
2. Watch this and fill in your lecture notes: Ch S3 Section 3.5.
POST-LEARNING:
Time to consolidate what you've learnt for the day: Consolidation Quiz.
Note: A copy of your responses for each Google Form submission will be sent to your email. Click View Score (for quiz) to review your responses and feedback again.
Page updated
Report abuse