| 1. |
Let X = the time it takes a read/write head to locate a desired record on a computer disk memory device once the head has been positioned over the correct track. If the disks rotate once every 30 millisec, a reasonable assumption is that X is uniformly distributed on the interval [0,30]. (20%)
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(a)Compute P(10≦X≦20). | (b) Compute P(X≧15). |
| (c)Obtain the cdf F(x). | (d) Compute E(X) and standard deviation.
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| 2. |
Bag 1 contains 3 red balls and 7 blue balls. Bag 2 contains 8 red balls and 4 blue
balls. Bag 3 contains 5 red balls and 11 blue balls. A bag is chosen at random, with
each bag being equally likely to be chosen, and then a ball is chosen at random
from that bag. Calculate the probability that
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(a) | a blue ball is chosen; |
| (b) | a red ball from Bag 2 is chosen; |
| (c) | If it is known that a blue ball is chosen, what is the probability that it comes from Bag 2? (15%) |
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| 3. |
Suppose that the random variables X and Y have a joint probability density function
f(x,y) = 4x(2-y)
for 0≦x≦1 and 1≦y≦2. (15%)
| (a) | What is the marginal probability density function of X? |
| (b) | Are the random variables X and Y independent? |
| (c) | What is the probability density function of X conditional on Y = 1.5? |
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| 4. |
Let  , and define a transformation T :R2 → R3
by T(x)=Ax, so that T(x)=Ax= (20%)
| (a) | Find T(u) , the image of u under the transformation T. |
| (b) | Find an x in R 2 whose image under T is b. |
| (c) | Is there more than one x whose image under T is b ? |
| (d) | Determine if c is in the range of the transmation T.
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| 5. |
Let  . Then {x 1 , x 2 , x 3 } is clearly linearly independent and thus is a basis for a subspace W of R 4 . Construct an orthogonal basis for W. (15%)
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| 6. |
Diagonalize the following matrix, if possible.
That is, find an invertible matrix P and a diagonal matrix D such that A = PDP -1 (15%) |