Answer :
The mean and variance of the reward of the runner-up is 7/3 and Variance is 89/9
What is Variance?
Variance is the anticipated squared variation of a random variable from its population mean or sample mean in probability theory and statistics. Variance serves as a proxy for dispersion, or the degree to which a set of numbers deviates from their mean.
Here, a=0 and b=12. So,
f(x) = 1 / (12 - 0)
= 1/12
A runner receives $10 if he or she is between 0 and 1 minute behind the leader. The corresponding probability is stated as
[tex]\int\limits^1_0 {1/12} \, dx[/tex]=1/12([tex]x^1_{0}[/tex])
= 1/12
A runner receives $6 if he or she is between one and three minutes behind the leader. The corresponding probability is stated as
[tex]\int\limits^3_1 {1/12} \, dx[/tex] = 1/12([tex]x^3_{1}[/tex])
= 1/6
A runner receives $2 if he or she is between three and six minutes behind the leader. The corresponding probability is stated as
[tex]\int\limits^6_3 {1/12} \, dx[/tex] = 1/12([tex]x^6_{3}[/tex])
= 1/4
A runner receives nothing if he or she is more than six minutes behind the leader. The corresponding probability is stated as
12
∫ 1/12 dx = 1/12 ([tex]x^{12}_6[/tex])
6
= 1/2
We can produce the following table:
X P(X) X(P(X) X^2 P(X)
10 1/12 5/6 25\3
6 1/6 1 6
2 1/4 1/2 1
0 1/2 0 0
Sum: 1 E(X)=7/3 E(X^2)=46/3
Thus the expected value of X is given as
E(X) = 7/3
The variance is given as
σ² = E(X)² - [E(X)]²
= 46/3 - (7/3)²
σ² = 89/9
Hence, Mean of the reward runner up is 7/3 and Variance is 89/9
To learn more about Variance click on the link
https://brainly.com/question/9304306
#SPJ4
