Probability Distributions

Probability Histogram: Graph of a Probability Distribution

It is similar to a relative frequency histogram, but the vertical scale shows probabilities instead of relative frequencies based on actual sample results.

Parameters of a Probability Distribution

With a probability distribution, we have a description of a population instead of a sample, so the values of the mean, standard deviation, and variance are parameters, not statistics.

Continuous Random Variable

A continuous random variable has infinitely many values, and the collection of values is not countable. That is, it is impossible to count the individual items because at least some of them are on a continuous scale, such as body temperatures.

Round-Off Rule for Mean, Variance, & Standard Deviation from a Probability Distribution

Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round the mean, variance, and standard deviation to one decimal place.

Probability Distribution Requirements

1. There is a numerical (not categorical) random variable x, and its number values are associated with corresponding probabilities. 2. The sum of all probabilities must be 1, but sums such as 0.999 or 1.001 are acceptable because they result from rounding errors. 3. Each probability value must be between 0 and 1 inclusive.

Discrete Random Variable

A discrete random variable has a collection of values that is finite or countable. If there are infinitely many value, the number of values is countable if it is possible to count them individually, such as the number of tosses of a coin before getting heads.

Probability Distribution

A probability distribution is a description that gives the probability of each value of the random variable. It is often expressed in the format of a table, formula, or graph.

Random Variable

A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure.

Expected Values Do Not have to be Whole Numbers

An expected value need not be a whole number, even if the different possible values of x might all be whole numbers. The expected number of girls in five births is 2.5, even though five particular births can never result in 2.5 girls. If we were to survey many couples with five children, we expect that the mean number of girls will be 2.5

Rare Event Rule for Inferential Statistics

If, under a given assumption, the probability of a particular outcomes is very small and the outcome occurs significantly less than or significantly greater than what we expect with the assumption, we conclude that the assumption is probably not correct.

Exceptions to Round-Off Rule

In some special cases, the round-off rule results in values that are misleading or inappropriate. For example, with four-engine jets the mean number of jet engines working successfully throughout a flight is 3.999714286, which becomes 4.0 when rounded, but that is misleading because it suggests that all jet engines always work successfully. Here we need more precision to correctly reflect the true mean, such as the precision in 3.999714.

Notation for 0+

In some tables and binomial probabilities, we sometimes use 0+ to represent a probability value that is positive but very small, such as 0.000000123. When rounding a probability value for inclusion in such a table, rounding to 0 would be misleading because it would incorrectly suggest that the event is impossible.

Identifying Significant Results with Probabilities

Significantly high numbers of successes: x successes among n trials is a significantly high number of successes if the probability of x or more successes is 0.05 or less. That is, x is a significantly high number of successes if P(x or more) is less than or greater than 0.05 Significantly low number of successes: x successes among n trials is a significantly low number of successes if the probability of x or fewer successes is 0.05 or less. That is, x is a significantly low number of success if P(x or fewer) is less than or greater than 0.05

Expected Value

The expected value of a discrete random variable x is denoted by E, and it is the mean value of the outcomes, so E = mean and E can can also be found by evaluating

Identifying Significant Results with the Range Rule of Thumb

The range rule of thumb may be helpful in interpreting the value of a standard deviation. According to the range rule of thumb, the vast majority of values should lie within 2 standard deviations of the mean, so we can consider a value to be significant if it is at least 2 standard deviations way from the mean.