ELEC 321
# Tutorial 3

Updated 2017-09-29

Given a random variable , and

The weights can be described as

so

and

**Consider**

are independent, , and

Thus and .

If . then , and .

We want to prove the following inequality:

Proof:

Letâ€™s start with definition of variance for continuous random variable:

**Consider** previous example:

Example: Simple coin tossExamining the expected values we get

So

The

`MATLAB`

code for this experiment is`clear p = rand(1); N = 1000; S = zeros(1, N+1) X_bar = zeros(1, N+1) for n = 1:N X_n = rand(1) < p; S(n + 1) = S(n) + X_i; X_bar(n + 1) = S(n + 1) / n; end`

Over time,

`X_bar`

will converge to a value since`p`

is a random number

For a majority decoding algorithm, if majority of the () transmitted identical digits are received correctly, then the received digit is considered correctly decoded. Let be the number of errors in the transmission of the () transmitted identical digits, and as the probability that each of the () bits can be decoded correctly on its own. Assume that the errors in each of the () positions are independent of each other.

So if the transmitted bits are

`000`

, for the receiving bits to be`010`

, the probability is

(a) If , what is the probability that one transmitted bit using majority decoding algorithm is decoded correctly?

Total bits are bits, so if there is at least 3 changed, it cannot be decoded correctly.

Then we can say the probability of 3 bits unchanged is

Which is the binomial random variable formula / distribution:

So the probability of decoding correctly is

Plugging the number in, we get

(b) If we only want use 3 identical bits in the majority decoding algorithm, what is the minimum required to have a better performance compared to (a)?

(c) If we use 7 identical bits, repeat (b).