Business Statistics - Chapter 6

Continuous random variable

A random variable that can assume any value in an interval on the real line or in a collection of intervals. Unlike discrete random variables, you cannot talk about the probability of it assuming one particular value; instead, you talk about the probability of it falling within a given interval.

Probability density function (PDF)

The function f(x) that describes the probability distribution of a continuous random variable. It does not give probabilities directly; instead, the area under the graph of f(x) over a given interval provides the probability that the variable falls within that interval.

Area under the curve equals probability

For any continuous random variable, the probability that x takes a value between a lower value x1 and a higher value x2 is found by computing the area under the graph of f(x) over the interval from x1 to x2. This principle is valid for all continuous distributions.

Probability of a single point is zero for a continuous random variable

Because a single point is an interval of zero width, the area under the curve at any single value is zero. The probability of a continuous random variable assuming any one exact value is always zero, and probabilities are the same whether or not the interval endpoints are included.

Total area under a probability density function equals one

The entire area under the graph of f(x) must equal 1, and f(x) must be greater than or equal to zero for all values of x. These requirements are the continuous analogs of the discrete rules that probabilities must sum to 1 and each probability must be non-negative.

Key difference between discrete and continuous probability distributions

For a discrete random variable, the probability function f(x) directly provides the probability of each value. For a continuous random variable, the probability density function f(x) does not give probabilities directly; probabilities are obtained only as areas under the curve of f(x) over an interval.

Uniform probability distribution

A continuous probability distribution in which the probability is proportional to the length of the interval. The random variable is equally likely to fall anywhere between the minimum value a and the maximum value b, making every sub-interval of equal length equally likely.

Uniform probability density function

Defined as f(x) = 1/(b minus a) for a less than or equal to x less than or equal to b, and f(x) = 0 elsewhere, where a is the smallest value and b is the largest value the variable can assume. The graph is a rectangle with constant height 1/(b minus a).

Expected value and variance of a uniform distribution

The expected value (mean) is E(x) = (a + b)/2, which is the midpoint of the interval. The variance is Var(x) = (b minus a) squared / 12. The standard deviation is the square root of the variance.

Computing probability for a uniform distribution

Because the density function has a constant height of 1/(b minus a), the probability of falling within any sub-interval equals the width of that sub-interval multiplied by the height. This is simply the area of the resulting rectangle.

The height of a probability density function is not itself a probability

The value of f(x) can exceed 1. For example, a uniform distribution on the interval 0 to 0.5 has f(x) = 2, which is greater than 1. Probability comes only from the area under the curve, not from the height of the function at a single point.

Normal probability distribution

The most commonly used continuous probability distribution, producing the familiar bell-shaped curve. It is widely used in statistical inference and in practical applications involving heights, weights, test scores, rainfall amounts, and scientific measurements. Abraham de Moivre published its derivation in 1733.

Parameters of the normal distribution

The entire family of normal distributions is defined by two parameters: the mean mu, which determines the center or location, and the standard deviation sigma, which determines the width and shape. Different values of mu shift the curve along the horizontal axis; different values of sigma change how flat or peaked the curve is.

Symmetry of the normal distribution

The normal distribution is perfectly symmetric around its mean, so the shape to the left of the mean is a mirror image of the shape to the right. Its skewness measure is zero. The tails extend to infinity in both directions without ever touching the horizontal axis.

Mean, median, and mode of a normal distribution

For a normal distribution, the mean, median, and mode are all equal. The highest point on the bell-shaped curve occurs at the mean. The mean can be any numerical value: negative, zero, or positive.

Effect of standard deviation on the normal curve

The standard deviation determines how flat and wide the normal curve is. Larger values of sigma produce wider, flatter curves showing more variability. Smaller values of sigma produce taller, narrower curves showing less variability.

Areas under the normal curve

The total area under the normal curve equals 1. Because the distribution is symmetric, exactly 0.50 (50 percent) of the area lies to the left of the mean and 0.50 lies to the right of the mean.

Empirical rule for normal distributions

Approximately 68.3 percent of values fall within plus or minus 1 standard deviation of the mean, approximately 95.4 percent fall within plus or minus 2 standard deviations, and approximately 99.7 percent fall within plus or minus 3 standard deviations. These percentages form the basis of the empirical rule.

Standard normal probability distribution

A special case of the normal distribution with a mean of 0 and a standard deviation of 1. The letter z is used to designate the standard normal random variable. It has the same bell shape as other normal distributions but is centered at zero with unit spread.

Converting to the standard normal distribution

Any normal random variable x with mean mu and standard deviation sigma can be converted to the standard normal variable z using z = (x minus mu) / sigma. The z value tells how many standard deviations x is from its mean. This conversion allows all normal probability questions to be answered using a single standard normal table or function.

Three types of standard normal probability calculations

(1) Cumulative (left-tail) probability: P(z less than or equal to a) is read directly from the table or computed with NORM.S.DIST. (2) Interval probability: P(a less than or equal to z less than or equal to b) = P(z less than or equal to b) minus P(z less than or equal to a). (3) Upper-tail probability: P(z greater than or equal to a) = 1 minus P(z less than or equal to a).

Using the standard normal probability table

The table provides the cumulative area under the standard normal curve to the left of a given z value. To look up z = 1.25, find the row labeled 1.2 and the column labeled 0.05; the intersection gives the cumulative probability. Negative z values appear on a separate page of the table.

Finding an x value given a probability (inverse normal)

When a desired cumulative probability is known, use the standard normal table in reverse to find the corresponding z value. Then convert back to x using x = mu + z times sigma. This technique is used for setting guarantees, thresholds, or cutoff values in applied problems.

NORM.S.DIST Excel function

Computes the cumulative probability (area to the left) for a given z value in the standard normal distribution. It takes two inputs: the z value and TRUE for cumulative probability or FALSE for the height of the curve. The S in the name stands for Standard normal.

NORM.S.INV Excel function

The inverse of NORM.S.DIST. It takes a cumulative probability as input and returns the z value that corresponds to that cumulative probability. Used when you know the desired area and need to find the z cutoff.

NORM.DIST Excel function

Computes the cumulative probability for any normal distribution, not just the standard normal. Its inputs are the x value, the mean mu, the standard deviation sigma, and TRUE for cumulative probability. It eliminates the need to convert x to z manually.

NORM.INV Excel function

The inverse of NORM.DIST. It takes a cumulative probability, the mean mu, and the standard deviation sigma as inputs and returns the x value corresponding to that cumulative probability. Useful for finding thresholds or guaranteed values directly without z conversion.

Computing interval and upper-tail probabilities in Excel for normal distributions

For an interval probability, subtract the NORM.S.DIST or NORM.DIST value at the lower endpoint from the value at the upper endpoint. For an upper-tail probability, subtract the cumulative probability from 1.

Exponential probability distribution

A continuous probability distribution useful for describing the time it takes to complete a task or the time between events. Common applications include time between vehicle arrivals at a toll booth, time required to complete a questionnaire, distance between major defects in a highway, and service times in waiting-line models.

Mean and standard deviation of the exponential distribution are equal

A distinctive property of the exponential distribution is that its mean and its standard deviation have the same value, so sigma = mu. The variance is therefore sigma squared = mu squared.

Skewness of the exponential distribution

The exponential distribution is skewed to the right with a skewness measure of 2. Its shape shows a high peak near zero that decreases rapidly, producing a long right tail. This is a clear example of a positively skewed distribution.

Cumulative probability for the exponential distribution

The probability that the exponential random variable x is less than or equal to a specific value x0 is P(x less than or equal to x0) = 1 minus e raised to the power of negative x0 divided by mu, where mu is the mean and e is approximately 2.71828. Interval and upper-tail probabilities are found by subtraction, just as with normal distributions.

Relationship between the Poisson and exponential distributions

If the number of occurrences in a given interval follows a Poisson distribution, then the length of the interval between consecutive occurrences follows an exponential distribution. If the Poisson mean is lambda occurrences per time period, then the exponential mean is mu = 1/lambda time periods per occurrence.

EXPON.DIST Excel function

Computes exponential probabilities. It takes three inputs: the x value, the rate parameter 1/mu (the inverse of the mean), and TRUE for cumulative probability or FALSE for the density function height. Interval probabilities use subtraction of two cumulative values; upper-tail probabilities subtract the cumulative result from 1.

Role of probability distributions in decision making

Once a probability distribution is established for a particular application, it can be used to obtain probability information about the problem. Probability does not make a decision recommendation directly, but it provides information that helps the decision maker better understand the risks and uncertainties, ultimately assisting in reaching a good decision.