Chapter 6 - Continuous Probability Distributions p.271

Question 1

Continuity correction factor p.290

Answer 1

A value of .5 that is added to or subtracted from a value of x when the continuous normal distribution is used to approximate the discrete binomial distribution.

Question 2

Exponential probability distribution p.293

Answer 2

A continuous probability distribution that is useful in computing probabilities for the time it takes to complete a task.

Question 3

Normal probability distribution p277

Answer 3

A continuous probability distribution. Its probability density function is bell-shaped and determined by its mean u and standard deviation o.

Question 4

Probability density function p.273

Answer 4

• A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.
• The counterpart of the probability function
  - The difference is that the probability density function does not directly provide probabilities. However, the area under the graph of f(x) corresponding to a given interval does provide the probability that the continuous random variable x assumes a value in that interval.
  - When we compute probabilities for continuous random variables we are computing the probability that the random variable assumes any value in an interval.

Question 5

Standard normal probability distribution p.279

Answer 5

A normal distribution with a mean of zero and a standard deviation of one.

Question 6

Uniform probability distribution p.273

Answer 6

A continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length.

Question 7

6.1 Uniform Probability Density Function

Answer 7

f(x) = 1/(b - a), for a <= x <= b

6.2 Normal Probability Density Functionf(x) = 0, elsewhere

Question 8

6.2 Normal Probability Density Function

Answer 8

§ f(x) = (1/(o * SquareRoot(2 * pie))) * e(-1/2)((x-u)/o)^2

• Where
  - U = mean
  - O = standard deviation
  - Pie = 3.14159
  - E = 2.71828

Question 9

6.3 Converting to the Standard Normal Random Variable

Answer 9

• The formula used to convert any normal random variable x with mean u and standard deviation o to the standard normal random variable z
  - z = (x - u) / o
• We answer probability questions about the distribution by first converting to the standard normal distribution. Then we can use the standard normal probability table and the appropriate z values to find the desired probability.

Question 10

6.4 Exponential Probability Density Function

Answer 10

f(x) = (1/u) * e^(-x/u), for x >= 0

• Where
  - U = expected value or mean

Question 11

6.5 Exponential Distribution: Cumulative Probabilities

Answer 11

To compute exponential probabilities

P(x <= x0) = 1 - e-x0/u

Question 12

Characteristics of the normal distribution

Answer 12

1. The entire family of normal distributions is differentiated by two parameters: the mean u and the standard deviation o.
2. The highest point on the normal curve is at the mean, which is also the median and mode of the distribution.
3. The mean of the distribution can be any numerical value: negative, zero, or positive.
4. The normal distribution is symmetric, with the shape of the normal curve to the left of the mean a mirror image of the shape of the normal curve to the right of the mean. The tails of the normal curve extend to infinity in both directions and theoretically not never touch the horizontal axis. Because it is symmetric, the normal distribution is not skewed; its skewness measure is zero
5. The standard deviation result in wider, flatter curves, showing more variability in the data
6. Probabilities for the normal random variable are given by areas under the normal curve. The total area under the curve for the normal distribution is 1. Because the distribution is symmetric, the area under the curve to the left of the mean is .50 and the area under the curve to the right of the mean is .
7. The percentage of values in some commonly used intervals are
   a. 68.3% of the values of a normal random variable are within plus or minus one standard deviation of its mean.
   b. 95.4% of the values of a normal random variable are within plus or minus two standard deviations of its mean.
   c. 99.7% of the values of a normal random variable are within plus or minus three standard deviation of its mean.

Question 13

Standard normal density function

Answer 13

f(z) = (1/(SqRt(2pie)) * e^(-(z^2)/2)