Question

Stereo speakers are manufactured with a probability of 0.100.10 being defective. TwentyTwenty speakers are randomly selected. Let the random variable X be defined as the number of defective speakers. Find the expected value and the standard deviation.

Answer 1

The expected value of X is 2 with a standard deviation of 1.34.

Explanation:

For each speaker, there are only two possible outcomes. Either it is defective, or it is not. The probability of a speaker being defective is independent of other speakers. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Stereo speakers are manufactured with a probability of 0.1 of being defective

This means that
`p = 0.1`

Twenty speakers are randomly selected.

This means that
`n = 20`

Let the random variable X be defined as the number of defective speakers. Find the expected value and the standard deviation.

`E(X) = np = 20*0.1 = 2`

`√(V(X)) = √(np(1-p)) = √(20*0.1*0.9) = 1.34`

The expected value of X is 2 with a standard deviation of 1.34.