Interpret the key results for Probability Distributions

Select the probability function that you want to interpret.

In This Topic

Probability density function (PDF)
Cumulative distribution function (CDF)
Inverse cumulative distribution function (ICDF)

Probability density function (PDF)

The probability density function helps identify regions of higher and lower probabilities for values of a random variable.

For a continuous distribution, Minitab calculates the probability density values.

Normal with mean = 0 and standard deviation = 1

x	f( x )
-3	0.004432
-2	0.053991
-1	0.241971
0	0.398942
1	0.241971
2	0.053991
3	0.004432

Key Results: x and f(x) for a continuous distribution

In these results, the probability density function is given for a normal distribution with mean = 0 and standard deviation = 1. For example, the function has a value of 0.00432 when the x-value is −3 or 3. The function has a value of 0.398942 when the x-value is 0.

For a discrete distribution, Minitab calculates the probability values. These values are also known as the probability mass function (PMF).

Binomial with n = 4 and p = 0.1

x	P( X = x )
0	0.6561
1	0.2916
2	0.0486
3	0.0036
4	0.0001

Key Results: x and P(X = x) for a discrete distribution

In these results, the probability density values are given for a binomial distribution with 4 trials and event probability of 0.10. For example, the probability that one event occurs in 4 trials is 0.2916, and the probability that 4 events occur in 4 trials is 0.0001.

Cumulative distribution function (CDF)

The cumulative distribution function (CDF) calculates the cumulative probability for a given x-value. Use the CDF to determine the probability that a data value is less than or equal to a certain value, higher than a certain value, or between two values.

For a continuous distribution, Minitab calculates the area under the probability density function, up to an x-value that you specify.

Normal with mean = 12 and standard deviation = 0.25

x	P( X ≤ x )
11.5	0.022750
12.5	0.977250

Key Results: x and P(X ≤ x) for a continuous distribution

In these results, suppose you assume that bottle fill weights are normally distributed with a mean of 12 ounces and a standard deviation of 0.25. The cumulative probability that a randomly chosen bottle has a fill weight that is less than or equal to 11.5 ounces is 0.022750. The cumulative probability that a randomly chosen bottle has a fill weight that is less than or equal to 12.5 ounces is 0.977250.

For a discrete distribution, Minitab calculates the cumulative probability values for the x-values that you specify.

Discrete uniform 1 to 6

x	P( X ≤ x )
1	0.16667
2	0.33333
3	0.50000
4	0.66667
5	0.83333
6	1.00000

Key Results: x and P(X ≤ x) for a discrete distribution

In these results, suppose you assume that you roll a fair die. You have a discrete integer probability of 1/6 for rolling each of the sides (1–6). The cumulative probability that you roll a 3 or less is 0.50000. The cumulative probability that you roll a 4 or less is 0.66667. The cumulative probability that you roll a 6 or less is 1.00000.

Inverse cumulative distribution function (ICDF)

The inverse cumulative distribution function (ICDF) provides the x-value for a specific cumulative probability.

For a continuous distribution, Minitab calculates the x-values for each cumulative probability that you specify.

Normal with mean = 1000 and standard deviation = 300

P( X ≤ x )	x
0.050	506.54
0.950	1493.46
0.025	412.01
0.975	1587.99

Key Results: P(X ≤ x) and x for a continuous distribution

In these results, the time by which 5% of the heating elements are expected to fail is the ICDF of 0.05, or approximately 507 hours. The time at which only 5% of the heating elements are expected to continue to function is the ICDF of 0.95, or approximately 1493 hours. The times between which the middle 95% of all heating elements are expected to fail is the ICDF of 0.025 and the ICDF of 0.975, or between approximately 412 and 1588 hours.

For a discrete distribution, an exact x-value for the cumulative probability that you specify might not exist. Therefore, Minitab displays exact integer values for the cumulative probabilities that are closest to the cumulative probability that you specify.

Binomial with n = 100 and p = 0.03

x	P( X ≤ x )	x	P( X ≤ x )
2	0.419775	3	0.647249

Key Results: P(X ≤ x) and x for a discrete distribution

In these results, the x-values are given for a binomial distribution with 100 trials and event probability of 0.03. For example, suppose you want to know the number of defectives that are associated with a cumulative probability of 50%. The cumulative probability is 0.419775 at x = 2, and the cumulative probability is 0.647249 at x = 3. The binomial distribution is a discrete distribution that cannot take x-values between 2 and 3, so no x-value for the exact cumulative probability of 0.50 exists.