Abel and Baker play a game of stubs the scores range from 0

Abel and Baker play a game of stubs the scores range from 0 to 120. Abels scores is N (70, 10) Baker\'s scores is N (60, 15) What is the joint probability density function? Use the joint probability density function f(x, y) = ? to answer: Probability that Abel\'s scores is between 50 and 80? Same but remove the scoring range condition? Probability that Baker\'s Scores 3 more than 4/5 times Abel\'s scores?

Solution

Let X be the Abel\'s score and Y be the Baker\'s score.

X ~ Normal (mean=70 , variance = 10)

Y ~ Normal (mean = 60, variance=15)

P(Abel\'s score between 50 and 80)

that is we have to find P(50 < X < 80)

Convert x=50 and x=80 into z-score.

z=(x-mean)/sd

sd = sqrt(variance) = sqrt(10) = 3.1623

z = (50 - 70) / 3.1623 = -6.3245

z = (80 - 70) / 3.1623 = 3.1623

That is now we have to find P(-6.3245 < Z < 3.1623)

= P(Z < 3.1623) - P(Z < -6.3245)

These probabilities we can find by using EXCEL.

syntax :

=normsdist(z)

where z is the z-score.

P(Z < 3.1623) = 0.9992

P(Z < -6.3245) =0.0000

P(-6.3245 < Z < 3.1623) = P(Z < 3.1623) - P(Z < -6.3245) = 0.9992 - 0.0000 = 0.9992