EL6303 Solution to HW 3 Fall 2015

1. Let (a) Find A so that represents a probability mass function. (b) Find (c) Find (d) Find the conditional probability mass function .(Video is required.)

Solution: Let’s discuss a more general problem.

1

Therefore, in our problem, (a)

(b)

(c)

(d)

2

x1 2

x

( )Xf x1/3 1

1 2

( )XF x

2. is uniform in [-1, 2] and .

(a) Find and draw . (b) Find and draw .

(c) Find and draw . (d) Find .(Video is required.)

Solution:

(a)

(b) & (c)

3

x

2y x

1 21x

( )Xf x1/3

1

4

1 2

4

That is

And

5

y

( )Yf y

41

16 y

13 y

1/31/ 6

1/12

y

1

( )YF y1

2/323

y1( 1)3

y

4

(d)

Check: . You check the details.

6

1( ) XF x

1

x

3. . And . Then (1) Find A so that f(x) is a valid probability density function.

(2) Find and draw .(3) Find .

(4) Find . Solution: (1) Clearly, A=1. (2)

(3) because for .Or, think this way:

7

la

(4) .4. A fine needle of length is dropped at random on a board covered with parallel lines distance apart where as in the figure.

(1) Find the probability that the needle intersects any of the lines. (2) Find the probability that the needle vertically intersects any of the lines. (3) Find the probability that the needle intersects a line with an angle between

needle and line is within ～ or ～ .

Solution:

8

a

Xl

/ 2a( / 2)sina

/ 2a

2/l x

(1) Let be the distance between the central point of the needle and the nearest line, and be the angle between the needle and the line counter-clockwise.Clearly, In order to intersect a line, we must have

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(2) P(the needle vertically intersects any of the lines)=0. (3) P(The needle intersects a line with an angle between needle and line is

within ～ or ～ )

5. ～(1) Find . (2) Find E{ }. (3) Find the second moment of Y.Solution:

(1)

(2)

------- (*)

Let

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(*)

(3) Similarly, .

6. Randomly pick a number from ( , , , ). Find the probability that the picked number equals .

Solution: and

Or: and

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Therefore, the probability that the picked number equals is ½.7. Show that Solution:

8. Show that

Solution:

9. Find

12

34

3( )xf x

x

( )xF x2

y

x

(In this solution, we are going to apply some properties of impulse

functions: a) . b)

c) For any function we have which is known as the sampling property of the impulse function.)

Solution:

13

y

( )yF y1

3

14

x

( )yf y1/2

3

10. Show that if then . Solution:

15

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