1



A certain college gives aptitude tests in the science and the humanities to all entering freshmen. If X and Y are, respectively, the proportions of correct answers that a student gets on the tests in the two subjects, the joint probability distribution of these random variables can be approximated with the joint probability density

[pic]

What is the probability that a student will get a total score of less than 0.80 from both tests? (Or equivalently, what proportion of the students will get a total score of less than 0.80 from both tests?)

Solution:

P(X+Y10, as shown in the shaded area below.

Since the joint density is uniform, the fraction of area indicates the probability. The fraction is 302/2/(30*40)=3/8. So the probability that man has to wait longer than 10 minutes is 3/8.

4. (Extra Credit) For a random sample X1, X2, …, Xn from a normal population N(((, (2), it can be shown that the sample mean [pic]and the sample variance

[pic]

are indeed, independent. Please prove this for the simple case of n=2. That is, when we have a sample of size 2.

Proof:

When n=2, we have X1, X2 only and

[pic]

[pic]

Let’s consider the joint moment generating function of (X1-X2)/2 and (X1+X2)/2:

M([pic],[pic])= E[exp([pic])]

= E[exp([pic])]

= E[exp([pic])] E[exp([pic])]

= exp([pic])exp([pic])

=exp([pic]) (Roy, please delete the last square – my pc keeps on dying when I use the math editor –Wei)

Recall the joint mgf of a bivariate normal RV is

M([pic],[pic])= (exp[pic].

So (X1-X2)/2 and (X1+X2)/2 follow bivariate normal distribution and the correlation coefficient is 0, which implied that they are independent. Since S2 is a function of (X1-X2)/2, so it is of course independent from [pic] too.

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Y

40

30

X

10

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