These problems pertain to the subject of sums of independent random variables. S

Question

These problems pertain to the subject of sums of independent random variables. Specifically, you need to know the distributions of the sum of two independent normals, and the distribution of the sum of two independent Poissons.

(1) Historically, when taking tests, the scores of student A in Math are normal with mean 1000 points and standard deviation 100 points. Those in Language are independent of the ones in Math and they are normal with mean 500 and standard deviation 10. The scores of student B in Math are normal with mean 1200 points and standard deviation 20 points. Those in Language are independent of the ones in Math and they are normal with mean 400 points and standard deviation 20 points. An entrance criterion to a college is that a prospective student must have a combined score of Math and Language (the sum of the scores in those subjects) above 1650 in an entrance test. Which student is more likely to enter that college?

Explanation / Answer

Here student A:

subject maths - Mean = 1000 and std. dev. = 100 points

subject language - Mean = 500 and std. deviaion = 10

Mean of combined score = 1000 + 500 = 1500

Std. deviation of combined score = sqrt [1002 + 102 ] = 100.5

Here student B:

subject maths - Mean = 1200 and std. dev. = 20 points

subject language - Mean = 400 and std. deviaion = 20

Mean of combined score = 1200 +400 = 1600

Std. deviation of combined score = sqrt [202 + 202 ] = 28.28

Now,

For student A and student B, we have to the probability that they will score higher than 1650. Whichever probability is higher , that student will likely to enter that college.

for student A :

Pr(x> 1650; 1500; 100.49) =1 - Pr(x< 1650; 1500; 100.49)

Z = (1650 - 1500)/ 100.49 = 1.49

Pr(x> 1650; 1500; 100.49) =1 - Pr(x< 1650; 1500; 100.49) = 1 - Pr(Z > 1.49)

= 1 - 0.9322

= 0.0678

for student B :

Pr(x> 1650; 1600; 28.28) =1 - Pr(x< 1650; 1600; 28.28)

Z = (1650 - 1600)/28.28 = 1.768

Pr(x> 1650; 1600; 28.28) =1 - Pr(x< 1650; 1600; 28.28) = 1 - Pr(Z > 1.768)

= 1 - 0.9615

= 0.0385

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