ECE 503 – Homework 11 1. Let 2
. a) To what value does converge? | 0.1. b) Find the smallest such that ⟹|
| 0.005. c) Find the smallest such that ⟹|
| 10 . ⟹|
d) Find the smallest such that |
e) Find the smallest such that ⟹|
. 2. Repeat problem 1 for 1
0.1 . 10
12
3. Let 0
otherwise
To what value does converge? 4. Let 0,1 , , ,
, and let . a) Sketch as a function of . b) Find and sketch as a function of . c) Sketch as a function of . d) Find and sketch as a function of . e) Find as a function of and . 5. Let 0,1 , and . a) To what value does 0 converge? b) To what value does 0.5 converge? c) To what value does 0.9 converge? d) To what value does 0.99 converge? 6. In problem 5, let . Note: and are not given (and are not needed). a) Find ∶
→
. b) Find ∶
↛
. c) Does converge to everywhere? 4
1
2
2
7. Let 1,3 , and ln
otherwise
a) To what value does b) To what value does c) To what value does d) To what value does e) To what value does 1 converge? 1.5 converge? 2 converge? 2.5 converge? 3 converge? 8. In problem 7, let ln . ↛
. a) Find ∶
b) Does converge to everywhere? c) Let and ,
, and find d) Does ⟶ almost everywhere? e) Find and sketch . 1 ,1
9. Let ~
1,1 and ~
, . 1,2, ⋯. Let and be independent and let . a) Find and sketch as a function of . b) Find and sketch as a function of . c) Find and sketch as a function of . d) Find and sketch as a function of . | 0.25 e) Find |
| 0.25 f) Find |
| 0.25 g) Find |
|
| 0.25 h) Find | 0.25 converge? i) To what value does |
| 0.01 j) Find |
| 0.01 k) Find |
| 0.01 l) Find |
| 0.01 converge? m) To what value does |
|
n) Let 0. To what value does |
converge? o) Does ⟶ in probability? 10. Let be Gaussian with 5 and 9. Let . and let a) Find and sketch . b) Find and sketch . c) Find and sketch . | d) Find Ε |
|
|
e) Find Ε
| f) Find Ε |
| converge? g) To what value does Ε |
h) Does ⟶ in mean square? i) For what values of does ⟶ in mean? be discrete with 1
0
11. Let 1,2, … be independent and identically distributed (i.i.d.), each having a Cauchy ∝
distribution. That is, ∝
. √
Note: is even, so we will let a) Show that √ ∝
∝
for some constant ∝
0. Let ∑
and let 0. . Hint: Use characteristic functions. Note: From a), is Cauchy for all . In particular, does not converge in distribution to Gaussian as one might expect from the central limit theorem. b) Why does not converge to a Gaussian random variable ?