Question

1. Consider the following random variables based on a roll of a fair die,

(1) You get $0.5 multiplied by the outcome of the die
(2) You get $0 if the outcome is below or equal 3 and $1 otherwise
(3) You get $1 if the outcome is below or equal 3 and $0 otherwise
(4) You pay $1.5 if the outcome is below or equal 3 and get $1 multiplied the outcome otherwise

Die Outcome	Random Variable (1)	Random Variable (2)	Random Variable (3)	Random Variable (4)
1	$0.50	$0.00	$1.00	-$1.50
2	$1.00	$0.00	$1.00	-$1.50
3	$1.50	$0.00	$1.00	-$1.50
4	$2.00	$1.00	$0.00	$4.00
5	$2.50	$1.00	$0.00	$5.00
6	$3.00	$1.00	$0.00	$6.00

Which random variable(s) has/have the highest variance?

a. Random Variable 1
b. Random Variable 2
c. Random Variable 3
d. Random Variable 4
e. Random Variable 2 and 3

Answer 1

Variance :

X = Random Variable

Avg = Average

n = No of Outcome = 06

Random Variable 1

Average = Sum of Variable / 06 = ( 0.5 + 1 + 1.5 + 2 + 2.5 + 3) / 6 = 1.75

= 4.375 / 06

= 0.73

Random Variable 2

Average = Sum of Variable / 06 = ( 0 + 0 + 0 + 1 + 1 + 1) / 6 = 0.50

  

= 1.5/ 6
= 0.25

Random Variable 3

Average = Sum of Variable / 06 = (1 + 1 + 1 + 0 + 0 + 0  ) / 6 = 0.50

= 1.5/ 6
= 0.25

Random Variable 4

Average = Sum of Variable / 06 = (-1.5 -1.5 -1.5 + 4 + 5 + 6 ) / 6 = 1.75

= 65.375/ 6
= 10.90

So Variable 4 is Having maximum Variance (Ans D)