確率分布【マーケターのためのデータサイエンスの時間】

以上が「データサイエンス力」のNo.10の解説になります。

二項分布【マーケターのためのデータサイエンスの時間】

• データサイエンス

2021.07.01

2024.10.22

松葉駿平

English

Explain 5 or more characteristics of typical probability distributions

Probability distributions tell the likelihood (chance) of an event. Usually statisticians use the following notation to describe probability:

p(x) = the likelihood that a random variable takes a specific value of x.

Probability distributions are either continuous probability distributions or discrete probability distributions, depending on whether they define probabilities for continuous or discrete variables [1].

 Following are some characteristics of probability distributions [2]:

1. The sum of all possible values of a probability is equal to 1.
2. The probability range value must be always between 0 and 1. The lowest probability while 1 shows the highest probability value.
3. In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.
4. Discrete probability functions are also known as probability mass functions and can assume a discrete number of values. For example, coin tosses and counts of events are discrete functions.
5. Probability distributions describe the dispersion of the values of a random variable. Consequently, the kind of variable determines the type of probability distribution. For a single random variable, statisticians divide distributions into the following two types:

• Discrete probability distributions for discrete variables.
• Probability density functions for continuous variables.

Discrete probability distribution

A discrete distribution describes the probability of occurrence of each value of a discrete random variable. A discrete random variable is a random variable that has countable values, such as a list of non-negative integers for example number of customer complaints [1].

With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability. Thus, a discrete probability distribution is often presented in tabular form

A discrete probability distribution function has two characteristics:

• Each probability is between zero and one, inclusive (inclusive means to include zero and one).
• The sum of the probabilities is equal to one.

Continuous probability distribution

A continuous probability distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable.

Probabilities of continuous random variables (X) are defined as the area under the curve of its PDF. The characteristic and significance of PDF are [3]:

• For a continuous random variable, one must consider the probability that it lies in an interval. The importance of this result is that it tells us that, to find the probability, we need to find the area under the pdf on the given interval.
• The total area under the pdf equals 1. This result means that in order to approximate the probability that the random variable lies in a given interval, we just have to guess the fraction of the area under the pdf between the ends of the interval. This result provides another perspective on why pdfs cannot be negative, since if they were, a negative probability could be obtained, which is impossible (i.e., area cannot be negative).