Question

A sequence of 1 million iid symbols(+1 and +2), Xi, are transmitted through a channel and summed to produce a new random variable W. Assume that the probability of transmitting a +1 is 0.4. Show your work

a) Determine the expected value for W
b) Determine the variance of W

Answer 1

E(w) = 1600000

v(w) = 240000

Explanation:

given data

sequence = 1 million iid (+1 and +2)

probability of transmitting a +1 = 0.4

solution

sequence will be here as

P{Xi = k } = 0.4 for k = +1

0.6 for k = +2

and define is

x1 + x2 + ................ + X1000000

so for expected value for W

E(w) = E( x1 + x2 + ................ + X1000000 ) ......................1

as per the linear probability of expectation

E(w) = 1000000 ( 0.4 × 1 + 0.6 × 2)

E(w) = 1600000

and

for variance of W

v(w) = V ( x1 + x2 + ................ + X1000000 ) ..........................2

v(w) = V x1 + V x2 + ................ + V X1000000

here also same as that xi are i.e d so cov(xi, xj ) = 0 and i ≠ j

so

v(w) = 1000000 ( v(x) )

v(w) = 1000000 ( 0.24)

v(w) = 240000