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Chapter 3: Problem 57

The space shuttle flight control system called PASS (Primary Avionics Software Set) uses four independent computers working in parallel. At each critical step, the computers "vote" to determine the appropriate step. The probability that a computer will ask for a roll to the left when a roll to the right is appropriate is \(0.0001 .\) Let \(X\) denote the number of computers that vote for a left roll when a right roll is appropriate. What is the mean and variance of \(X ?\)

Short Answer

Expert verified
Mean: 0.0004, Variance: 0.00039996

Step by step solution

01

Identify the Type of Distribution

Since we have a fixed number of independent trials with only two possible outcomes (a roll to the left or not), we use the binomial distribution. Here, the number of trials \( n \) is 4 (as there are four computers), and the probability of success \( p \) (voting incorrectly) is 0.0001.
02

Calculate the Mean

The mean of a binomial distribution is given by the formula \( \mu = n \times p \). Substituting the values, we get:\[ \mu = 4 \times 0.0001 = 0.0004. \] Therefore, the mean number of computers that vote incorrectly is 0.0004.
03

Calculate the Variance

The variance of a binomial distribution is given by the formula \( \sigma^2 = n \times p \times (1 - p) \). Substituting the values, we get:\[ \sigma^2 = 4 \times 0.0001 \times (1 - 0.0001) = 4 \times 0.0001 \times 0.9999. \] Calculating further, \[ \sigma^2 = 0.00039996. \] Thus, the variance is 0.00039996.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean of Binomial Distribution
The mean of a binomial distribution is an essential concept in statistics that helps describe the expected outcome of a series of independent trials. When you have a set number of trials where there are only two possible outcomes, such as success and failure, the binomial distribution applies. In the context of our example with the space shuttle, the "success" is defined as a computer voting incorrectly for a left roll when a right roll is needed.
To calculate the mean of a binomial distribution, we use the formula:
  • \[\mu = n \times p \]
where \( n \) is the number of trials, and \( p \) is the probability of success in each trial. In the shuttle system, with four computers functioning as independent trials and a very low probability of voting error (0.0001), the expected number of computers making an incorrect vote (the mean) is just 0.0004. This indicates that on average, virtually no computer would vote incorrectly given the parameters.
Variance of Binomial Distribution
Variance in a binomial distribution measures how much the outcomes differ from the mean. It tells us how much we can expect the actual results to vary. When dealing with the binomial distribution, variance is an important measure of uncertainty and spread around the mean.
To calculate the variance, use the formula:
  • \[\sigma^2 = n \times p \times (1 - p)\]
where \( n \) is the number of trials, \( p \) is the probability of success, and \( 1 - p \) is the probability of failure. For the space shuttle's voting system, the variance calculated as 0.00039996 signifies that there is a very tiny expected deviation from the mean. In real-world terms, this means there's almost no variation expected in the number of incorrect votes.
Probability of Independent Trials
A crucial element in understanding the binomial distribution is the concept of independent trials. Each trial must be independent, meaning the outcome of one trial doesn鈥檛 affect the others. This is fundamental because the binomial model assumes that each attempt at "success" or "failure" is unaffected by previous attempts. In the case of the space shuttle, each computer operates independently when deciding its vote.
The probability of a specified outcome happening in a binomial distribution builds on this independence. If trials aren鈥檛 independent, the probability calculations wouldn't hold true. This independence allows us to confidently use the binomial formulas for mean and variance to describe the expected outcomes and their variability for multiple attempts, such as with the computers in the space shuttle example. With single probabilities multiplied across all trials, independence ensures results are mathematically accurate.

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Most popular questions from this chapter

Assume the number of errors along a magnetic recording surface is a Poisson random variable with a mean of one error every \(10^{5}\) bits. A sector of data consists of 4096 eight-bit bytes. (a) What is the probability of more than one error in a sector? (b) What is the mean number of sectors until an error is found?

Suppose that \(X\) has a discrete uniform distribution on the integers 0 through \(9 .\) Determine the mean, variance, and standard deviation of the random variable \(Y=5 X\) and com- pare to the corresponding results for \(X\).

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