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Random Variable

5.1 Random Variable
 
The image is an educational graphic about random variables. It has two main sections in cloud-shaped bubbles. The first section, labeled 'DEFINITION,' explains that a random variable is a variable with numeric values that can be determined from a random phenomenon. The second section, labeled 'NOTATION,' shows an example of notation for a random variable, X = {0, 1, 2}. The title 'RANDOM VARIABLE' is prominently displayed on the left side, with the logo 'Pandai' beneath it.
 
Two Types of Random Variables
Discrete Random Variable Continuous Random Variable
Random variables that have countable numbers of values, usually taking values like zero and positive integers. Random variables that are not integers but take values that lie in an interval.
 
Discrete Random Variable

If \(X\) represents a discrete random variable, hence the possible outcomes can be written in set notation:

\(X = \lbrace r : r = 0, 1, 2, 3\rbrace\)
 
Continuous Random Variable

If \(Y\) represents a continuous random variable, hence the possible outcomes can be written as:

\(Y = \lbrace y : y\) is the pupil's height in cm, \(a \leq y \leq b \rbrace\)
 
Probability Distribution for Discrete Random Variables
If \(X\) is a discrete random variable with the values \(r_1\), \(r_2\), \(r_3\), \(...\), \(r_n\) and their respective probabilities are \(P(X=r_1)\), \(P(X=r_2)\), \(P(X=r_3)\), \(...\), \(P(X=r_n)\), then \(\sum_{i=1}^{n} P(X=r_i)=1\), thus each \(P(X=r_i) \geq0\).
 
Example
Question

\(70\%\) of Form \(5\) Dahlia pupils achieved a grade A in the final year examination for the Science subject. Two pupils were chosen at random from that class.

(a) If \(X\) represents the number of pupils who did not get a grade A, construct a table to show all the possible values of \(X\) with their corresponding probabilities.
(b) Next, draw a graph to show the probability distribution of \(X\).
Solution (a)

\(P (A : A \text{ is a pupil who did not achieve a grade A}) \\ = 1-\dfrac{70}{100}\\ = 0.3.\)

\(P (B : B \text{ is a pupil who achieved a grade A}) \\ = \dfrac{70}{100}\\ = 0.7.\)

Then, \(X = \{0, 1, 2\}\).

\(\begin{aligned} P(X = 0) &= P(B, B) \\ &= 0.7 × 0.7\\ &=0.49.\\\\ P(X = 1) &= P(A, B) + P(B, A)\\ &= (0.3 × 0.7) + (0.7 × 0.3)\\ &=0.42.\\\\ P(X = 2) &= P(A, A) \\ &= 0.3 × 0.3\\ &=0.09 .\end{aligned}\)

\(X=r\) \(0\) \(1\) \(2\)
\(P(X=r)\) \(0.49\) \(0.42\) \(0.09\)
Solution (b)

A graph depicting the probability distribution of X, featuring a line and several plotted points illustrating data trends.
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.

Random Variable

5.1 Random Variable
 
The image is an educational graphic about random variables. It has two main sections in cloud-shaped bubbles. The first section, labeled 'DEFINITION,' explains that a random variable is a variable with numeric values that can be determined from a random phenomenon. The second section, labeled 'NOTATION,' shows an example of notation for a random variable, X = {0, 1, 2}. The title 'RANDOM VARIABLE' is prominently displayed on the left side, with the logo 'Pandai' beneath it.
 
Two Types of Random Variables
Discrete Random Variable Continuous Random Variable
Random variables that have countable numbers of values, usually taking values like zero and positive integers. Random variables that are not integers but take values that lie in an interval.
 
Discrete Random Variable

If \(X\) represents a discrete random variable, hence the possible outcomes can be written in set notation:

\(X = \lbrace r : r = 0, 1, 2, 3\rbrace\)
 
Continuous Random Variable

If \(Y\) represents a continuous random variable, hence the possible outcomes can be written as:

\(Y = \lbrace y : y\) is the pupil's height in cm, \(a \leq y \leq b \rbrace\)
 
Probability Distribution for Discrete Random Variables
If \(X\) is a discrete random variable with the values \(r_1\), \(r_2\), \(r_3\), \(...\), \(r_n\) and their respective probabilities are \(P(X=r_1)\), \(P(X=r_2)\), \(P(X=r_3)\), \(...\), \(P(X=r_n)\), then \(\sum_{i=1}^{n} P(X=r_i)=1\), thus each \(P(X=r_i) \geq0\).
 
Example
Question

\(70\%\) of Form \(5\) Dahlia pupils achieved a grade A in the final year examination for the Science subject. Two pupils were chosen at random from that class.

(a) If \(X\) represents the number of pupils who did not get a grade A, construct a table to show all the possible values of \(X\) with their corresponding probabilities.
(b) Next, draw a graph to show the probability distribution of \(X\).
Solution (a)

\(P (A : A \text{ is a pupil who did not achieve a grade A}) \\ = 1-\dfrac{70}{100}\\ = 0.3.\)

\(P (B : B \text{ is a pupil who achieved a grade A}) \\ = \dfrac{70}{100}\\ = 0.7.\)

Then, \(X = \{0, 1, 2\}\).

\(\begin{aligned} P(X = 0) &= P(B, B) \\ &= 0.7 × 0.7\\ &=0.49.\\\\ P(X = 1) &= P(A, B) + P(B, A)\\ &= (0.3 × 0.7) + (0.7 × 0.3)\\ &=0.42.\\\\ P(X = 2) &= P(A, A) \\ &= 0.3 × 0.3\\ &=0.09 .\end{aligned}\)

\(X=r\) \(0\) \(1\) \(2\)
\(P(X=r)\) \(0.49\) \(0.42\) \(0.09\)
Solution (b)

A graph depicting the probability distribution of X, featuring a line and several plotted points illustrating data trends.
 
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of pandai.
Chapter : Probability Distributions
Topic : Random Variable
Form 5 Additional Mathematics
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