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Continuous Random Variables and Their Probability Distributions Solved Problems for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Review the following concepts if necessary:

Uniform Probability Distribution

1. The price for a gallon of whole milk is uniformly distributed between $2.25 and $2.75 during July in the U.S. Give the equation and graph the pdf for X, the price per gallon of whole milk during July. Also determine the percent of stores that charge more than $2.70 per gallon.

Continuous Random Variables and Their Probability Distributions Solved Problems

The percent of stores charging a higher price than $2.70 is P(X > 2.70) times 100. The probability P(X > 2.70) is the shaded area in Fig. 6-35. This area is 2 × .05 = .10. Ten percent of all milk outlets sell a gallon of milk for more than $2.70.

2. The time between release from prison and the commission of another crime is uniformly distributed between 0 and 5 years for a high-risk group. Give the equation and graph the pdf for X, the time between release and the commission of another crime for this group. What percent of this group will commit another crime within two years of their release from prison?

The percent who commit another crime within two years is given by P(X < 2) times 100. This probability is shown as the shaded area in Fig. 6-37, and is equal to 2 × .2 = .4. Forty percent will commit another crime within two years.

Continuous Random Variables and Their Probability Distributions Solved Problems

Mean And Standard Deviation For The Uniform Probability Distribution

3. Find the mean and standard deviation of the milk prices in Problem 6.1. What percent of the prices are within one standard deviation of the mean?

Ans. The mean is given by and the standard deviation is given by The one standard deviation interval about the mean goes from 2.36 to 2.64 and the probability of the interval is (2.64 – 2.36) × 2 = .56. Fifty-six percent of the prices are within one standard deviation of the mean.

4. Find the mean and standard deviation of the times between release from prison and the commission of another crime in Problem 6.2. What percent of the times are within two standard deviations of the mean?

Ans. The mean is given by and the standard deviation is given by A 2 standard deviation interval about the mean goes from –.38 to 5.38 and 100% of the times are within 2 standard deviations of the mean.

Normal Probability Distribution

5. The mean net worth of all Hispanic individuals aged 51–61 in the U.S. is $80,000, and the standard deviation of the net worths of such individuals is $20,000. If the net worths are normally distributed, what percent have net worths between: (a) $60,000 and $100,000; (b) $40,000 and $120,000; (c) $20,000 and $140,000?

Ans.

  1. 68.26% have net worths between $60,000 and $100,000.
  2. 95.44% have net worths between $40,000 and $120,000.
  3. 99.72% have net worths between $20,000 and $140,000.

6. If the median amount of money that parents in the age group 51–61 gave a child in the last year is $1725 and the amount that parents in this age group give a child is normally distributed, what is the modal amount that parents in this age group give a child?

Ans. Since the distribution is normally distributed, the mean, median, and mode are all equal. Therefore, the modal amount is also $1725.

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