Random Variable

5.1

Random Variable

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.

Random Variable

5.1

Random Variable

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

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, \(X = \{0, 1, 2\}\).

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

Solution (b)

A graph depicting the probability distribution of X, featuring a line and several plotted points illustrating data trends.

Random Variable

Random Variable

Chapter : Probability Distributions

Topic : Random Variable

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