MaN)Random variable Review of probability and random variables Suppose that we transmit a 3 1-bit sequence with error correction capability up to 3 bit errors If the probability of a bit error is p=0.001, what is the probability that the sequence received is in error? P(sequence error )=1- P(correct sequence) 3 =1-∑ 2)(0.001)(0.99)31-≈310-8 31 On the other hand if no error correction is used the error probability is 1-(0.999)31≈0.0305 Communications Engineering
Communications Engineering 16 Random variable Review of probability and random variables ➢ Suppose that we transmit a 31-bit sequence with error correction capability up to 3 bit errors ➢ If the probability of a bit error is p=0.001, what is the probability that the sequence received is in error? ➢ On the other hand, if no error correction is used, the error probability is
MaN)Random variable Review of probability and random variables Uniform distribution (x) 1 f <a<b X()= b-a 0 otherwise The random phase of a sinusoid is often modeled as a uniform rv between0 and 2T The mean or expected value of X is mx=E[x]=∑xP first moment ofX Ix=E[x]= Communications Engineering
Communications Engineering 17 Random variable Review of probability and random variables ➢ Uniform distribution: ➢ The random phase of a sinusoid is often modeled as a uniform r.v. between 0 and ➢ The mean or expected value of X is first moment of X
MaN)Random variable Review of probability and random variables Then- th moment ofⅩ x"pr(xdx Ifn=2, we have the mean-squared value ofX E[x2]= x pr(xdx The n-th central moment is ELCX-mx)]=[(-myfr(x)dx >Ifn-2 we have the variance ofx 0=ECx-mx) >or is called the standard deviation X-2myx+m E|x2]-m2 Communications Engineering 18
Communications Engineering 18 Random variable Review of probability and random variables ➢ The n-th moment of X ➢ If n=2, we have the mean-squared value of X ➢ The n-th central moment is ➢ If n=2, we have the variance of X ➢ is called the standard deviation
MaN)Random variable Review of probability and random variables Gaussian distribution fr(r) fx(x)=_1 exp X A Gaussian r v. is completely determined by its mean and variance, and hence usually denoted as X-Nlmx,02) The most important distribution in communications. Communications Engineering 19
Communications Engineering 19 Random variable Review of probability and random variables ➢ Gaussian distribution: ➢ A Gaussian r.v. is completely determined by its mean and variance, and hence usually denoted as The most important distribution in communications!
MaN)Random variable Review of probability and random variables > Q-function is a standard form to express error probabilities without a closed-form exI 2丌 The Q-function is the area under the tail of a gaussian pdf with mean 0 and variance I N(0.1) O(x) 0 X Extremely important in error probability analysis Communications Engineering
Communications Engineering 20 Random variable Review of probability and random variables ➢ Q-function is a standard form to express error probabilities without a closed-form ➢ The Q-function is the area under the tail of a Gaussian pdf with mean 0 and variance 1 Extremely important in error probability analysis!