1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Bas_tet [7]
3 years ago
5

A source of information randomly generates symbols from a four letter alphabet {w, x, y, z }. The probability of each symbol is

as follows: P(w) = 1 2 ; P(x) = 1 4 ; P(y) = 1 8 ; and P(z) = 1 8 . These symbols are now encoded into binary codes using the scheme shown below. Let the random variable L denote the length of the binary code and pL(l) denote the PMF of L.symbol codew 0x 10y 110z 111Find the expectation and variance of L.
Mathematics
1 answer:
koban [17]3 years ago
5 0

The expected length of code for one encoded symbol is

\displaystyle\sum_{\alpha\in\{w,x,y,z\}}p_\alpha\ell_\alpha

where p_\alpha is the probability of picking the letter \alpha, and \ell_\alpha is the length of code needed to encode \alpha. p_\alpha is given to us, and we have

\begin{cases}\ell_w=1\\\ell_x=2\\\ell_y=\ell_z=3\end{cases}

so that we expect a contribution of

\dfrac12+\dfrac24+\dfrac{2\cdot3}8=\dfrac{11}8=1.375

bits to the code per encoded letter. For a string of length n, we would then expect E[L]=1.375n.

By definition of variance, we have

\mathrm{Var}[L]=E\left[(L-E[L])^2\right]=E[L^2]-E[L]^2

For a string consisting of one letter, we have

\displaystyle\sum_{\alpha\in\{w,x,y,z\}}p_\alpha{\ell_\alpha}^2=\dfrac12+\dfrac{2^2}4+\dfrac{2\cdot3^2}8=\dfrac{15}4

so that the variance for the length such a string is

\dfrac{15}4-\left(\dfrac{11}8\right)^2=\dfrac{119}{64}\approx1.859

"squared" bits per encoded letter. For a string of length n, we would get \mathrm{Var}[L]=1.859n.

You might be interested in
Jayson said, "I am thinking of two fractions that when added have a sum of 1." Which fractions could Jayson have been thinking a
viva [34]

Answer:

3/6 and 3/6

thanks mate

5 0
2 years ago
Read 2 more answers
Check my work?<br><br> Rational and irrational numbers.
Sauron [17]
Here, Rational numbers are: = 2/9 and √225
& other two are irrational

Hope this helps!
3 0
3 years ago
Read 2 more answers
The cost of a call was $1.00
baherus [9]
I want chicken nuggets
3 0
2 years ago
Half of farihas hockey cards got wet and ruined.fariha bought 5 cards to replace some that were lost
nadya68 [22]

There is approximately 10 cards at the start

8 0
3 years ago
-3q– 19 = 19 – 5q<br> q=
Sidana [21]

Answer:

19

Step-by-step explanation:

-3q+5q=38

2q=38

q=19

please like my answer

5 0
3 years ago
Read 2 more answers
Other questions:
  • A moving company charges $40 plus $0.25 per mile to rent a van. Another company charges $25 plus $0.35 per mile to rent the same
    12·1 answer
  • Select the correct answer.
    8·1 answer
  • The formula for the volume of a sphere is V = Ar3, where Vis the volume
    8·1 answer
  • Help please? Graphs just are not my thing.
    10·1 answer
  • Use base ten blocks to solve 53/4
    5·1 answer
  • Jerome scores 12 points in a basketball game.This is twice the number of points that Jaime scores how many points did the rest o
    15·2 answers
  • Janet invested $10,000, part at 2% and part at 12%. if the total interest at the end of the year is $600, how much did she inves
    9·1 answer
  • Help please asap. i’ll mark brainliest
    6·1 answer
  • A tank in the shape of an inverted right circular cone has height 7 meters and radius 2 meters. It is filled with 5 meters of ho
    6·2 answers
  • find the equation of the striaght line passing through (3, 3) which is perpendicular to the line y=-1/2x-4
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!