I mean statements such as: "the probability distribution of X" or "its probability distribution". I am quite confused about such expressions. Could someone please explain the relationship between, Probability, Probability Distribution, density function? I have checked on Wikipedia for a while, still only clear about that, density function can be used to calculate the probability.
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1$\begingroup$ Sometimes "probability distribution" is used as shorthand for "probability density function OR probability mass function", when $X$ could be discrete or continuous. See also this CV question. $\endgroup$– GeoMatt22Oct 25, 2016 at 3:47
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2 Answers
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The term probability relates is to an event and probability distribution relates is to a random variable.
It is a convention that the term probability mass function refers to the probability distribution of a discrete random variable and the term probability density function refers to the probability function of a continuous random variable.
When we say probability distribution it may pertain to a discrete random variable or a continuous random variable, depending on the context.
When the random variable is discrete, probability distribution means, how the total probability is distributed over various possible values of the random variable. Consider the experiment of tossing two unbiased coins simultaneously. Then, sample space $S$ associated with this experiment is: $$S=\left\lbrace HH, HT, TH, TT \right\rbrace$$
If we define a random variable $X$ as: the number of heads on this sample space $S$, then $X(HH)=2, X(HT)=X(TH)=1, X(TT)=0$. The probability distribution of $X$ is then given by
\begin{array}{cccc} X=x & 0 & 1 & 2\\ & & &\\ p(x)& \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \end{array}
For a discrete random variable, we consider events of the type $\{X=x\}$ and compute probabilities of such events to describe the distribution of the random variable. This probability, $P\{X=x\}=p(x)$ is called the probability mass function and it satisfies the properties:
\begin{eqnarray*}
0\leq p(x)\leq 1 \mbox{ and } \sum_{x}p(x)=1.
\end{eqnarray*}
A continuous random variable is described by considering probabilities for events of the type $(a<X<b)$. The probability function of a continuous random variable is usually denoted by $f(x)$ and is called a density function.
$$P(a<X<b)=\int_{a}^{b}f(x)dx$$
where $f(x)$ satisfies the conditions
\begin{equation*}
f(x)\geq 0, \mbox{ and } \int_{-\infty}^{\infty}f(x)dx = 1.
\end{equation*}
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Probability: is the likelihood of a possible outcome.
Probability Distribution: assigns a probability to each measurable subset of the possible outcomes of a random experiment.
Probability Density Function: Calculates the assigned probability of a particular subset of a Probability Distribution.
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1$\begingroup$ You might want to add how each relate back to the value 1. $\endgroup$– conv3dOct 25, 2016 at 3:06
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$\begingroup$ So is it correct to say that the term "probability distribution" has exactly the same meaning as the term "probability measure"? $\endgroup$– littleOJun 8, 2019 at 23:49
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$\begingroup$ @littleO Look here:(math.stackexchange.com/questions/384923/…) $\endgroup$ Jun 9, 2019 at 10:27
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1$\begingroup$ (-1) Not helpful, unfortunately. As likelihood is a term with its own strong and specific technical flavour, even using the word informally really doesn't help, especially if you are purporting to give definitions. (I corrected the spelling "likely hood" which is just wrong.) The definition of pdf here is somewhere between meaningless and wrong. $\endgroup$– Nick CoxApr 15, 2020 at 13:08