### Probability and Statistics: Exercises on Probability Distributions **Exercise 7.34** For a given computer salesperson, the probability distribution of \( x \), which is the number of systems sold in one month, is as follows: | \( x \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |-------|----|----|----|----|----|----|----|----| | \( p(x) \) | 0.05 | 0.10 | 0.12 | 0.30 | 0.30 | 0.11 | 0.01 | 0.01 | - **a.** Calculate the mean value of \( x \) (the mean number of systems sold). - **b.** Determine the variance and standard deviation of \( x \). Provide an interpretation of these values. - **c.** Calculate the probability that the number of systems sold is within 1 standard deviation of its mean value. - **d.** Calculate the probability that the number of systems sold is more than 2 standard deviations from the mean. **Exercise 7.35** A local television station offers advertising spots of different lengths: 15-second, 30-second, and 60-second. Let \( x \) represent the length of a randomly selected commercial. The probability distribution is: | \( x \) | 15 | 30 | 60 | |-------|-----|-----|-----| | \( p(x) \) | 0.1 | 0.3 | 0.6 | - **a.** Find the average length of randomly selected advertising spots. This exercise set aims to strengthen your understanding of probability distributions, particularly focusing on calculating statistical measures such as mean, variance, and standard deviation, and understanding their implications in real-world scenarios.
### Probability and Statistics: Exercises on Probability Distributions **Exercise 7.34** For a given computer salesperson, the probability distribution of \( x \), which is the number of systems sold in one month, is as follows: | \( x \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |-------|----|----|----|----|----|----|----|----| | \( p(x) \) | 0.05 | 0.10 | 0.12 | 0.30 | 0.30 | 0.11 | 0.01 | 0.01 | - **a.** Calculate the mean value of \( x \) (the mean number of systems sold). - **b.** Determine the variance and standard deviation of \( x \). Provide an interpretation of these values. - **c.** Calculate the probability that the number of systems sold is within 1 standard deviation of its mean value. - **d.** Calculate the probability that the number of systems sold is more than 2 standard deviations from the mean. **Exercise 7.35** A local television station offers advertising spots of different lengths: 15-second, 30-second, and 60-second. Let \( x \) represent the length of a randomly selected commercial. The probability distribution is: | \( x \) | 15 | 30 | 60 | |-------|-----|-----|-----| | \( p(x) \) | 0.1 | 0.3 | 0.6 | - **a.** Find the average length of randomly selected advertising spots. This exercise set aims to strengthen your understanding of probability distributions, particularly focusing on calculating statistical measures such as mean, variance, and standard deviation, and understanding their implications in real-world scenarios.
MATLAB: An Introduction with Applications
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Transcribed Image Text:### Probability and Statistics: Exercises on Probability Distributions
**Exercise 7.34**
For a given computer salesperson, the probability distribution of \( x \), which is the number of systems sold in one month, is as follows:
| \( x \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|-------|----|----|----|----|----|----|----|----|
| \( p(x) \) | 0.05 | 0.10 | 0.12 | 0.30 | 0.30 | 0.11 | 0.01 | 0.01 |
- **a.** Calculate the mean value of \( x \) (the mean number of systems sold).
- **b.** Determine the variance and standard deviation of \( x \). Provide an interpretation of these values.
- **c.** Calculate the probability that the number of systems sold is within 1 standard deviation of its mean value.
- **d.** Calculate the probability that the number of systems sold is more than 2 standard deviations from the mean.
**Exercise 7.35**
A local television station offers advertising spots of different lengths: 15-second, 30-second, and 60-second. Let \( x \) represent the length of a randomly selected commercial. The probability distribution is:
| \( x \) | 15 | 30 | 60 |
|-------|-----|-----|-----|
| \( p(x) \) | 0.1 | 0.3 | 0.6 |
- **a.** Find the average length of randomly selected advertising spots.
This exercise set aims to strengthen your understanding of probability distributions, particularly focusing on calculating statistical measures such as mean, variance, and standard deviation, and understanding their implications in real-world scenarios.
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