Example 4-18: Applying the Approximation In a digital...

Example 4-18: Applying the Approximation In a digital communication channel, assume that the number of bits received in error can be modeled by a binomial random variable. The probability that a bit is received in error is $10^{-5}$. If 16 million bits are transmitted, what is the probability that 150 or fewer errors occur? $P(X \leq 150) = P(X \leq 150.5)$

Transcript

00:01 For this question, let me take notes what is given here.

00:03 So the probability was given here, which was given as 10 to the power minus 5.

00:09 And the number of trial was given as 16 million.

00:13 16 million can be written as 16 times 10 to the bar bar of 6.

00:17 So i can define the random variable x, which is binomily distributed.

00:21 This is 16 times 10 to the power 6, and the probability is 10 to power minus 5.

00:26 So we can use the normal approximation here.

00:30 What we have to do, we need to check the n times p should be greater than 5.

00:34 And also n times 1 minus p should be greater than or equal to 5 here.

00:39 Let's check this one here.

00:41 This is 16, 10 to power 6 times 10 to power minus 5, which is equal to 160, which is greater than 5.

00:48 That works.

00:49 And for this one, this is 16, 10 to the power 6 and 1 minus 10 to power minus 5, which is also greater than 5.

00:56 So we can also use the normal approximation here.

01:00 So we need to get the mean value here.

01:02 The mean is n times p.

01:03 We got this value before, which is 160.

01:07 And the standard deviation of this distribution would be, which is the square of n times p times 1 minus p.

01:13 So this is equal to 16 times 10 to the power 6, 10 to the power negative 5, and 1 minus 10 to the power negative 5.

01:23 Let me just get the standard deviation, which is the square root of.

01:26 This is 16, n times 10 to the power 6...

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