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I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$.

Now, a probability density function of of a continuous random variable X is $y=f(x)$. Wikipedia defines this function $y$ to mean

In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

I am confused about the meaning of 'relative likelihood' because it certainly does not mean probability! The probability $P(X<x_0)$ is given by some integral of the pdf.

So what does $f(x_0)$ indicate? It gives a real number, but isn't the relative likelihood of a specific value for a CRV always zero?

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Let $f$ be the density function of $X$. Assume $f$ is continuous. Then if $h$ is small, the probability that $X$ lies in the interval $[a,a+h]$ is approximately $hf(a)$. By approximately I mean that the probability, divided by $h$, approaches $f(a)$ as $h$ approaches $0$. So the ratio $f(a)/f(b)$ measures, approximately, the ratio of the probability that $X$ is in $[a,a+h]$ to the probability $X$ is in $[b,b+h]$. – André Nicolas Oct 10 '12 at 19:28

3 Answers 3

up vote 3 down vote accepted

'Relative likelihood' is indeed misleading. Look at it as a limit instead: $$ f(x)=\lim_{h \to 0}\frac{F(x+h)-F(x)}{h} $$ where $F(x) = P(X \leq x)$

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So you suggest looking at the pdf as being defined by the cumulative distribution function? – jesterII Oct 10 '12 at 19:39
This is essentially the definition of pdf fro CRVs – Alex Oct 10 '12 at 20:12
A good way of thinking about is $f(x) = \frac{dF}{dx}$ and so it's the rate of change of the cdf at $x$. – Jacob Feb 27 '13 at 17:39
Why the downvote? What's wrong with what I wrote? – Alex Feb 27 '13 at 17:48

In general, if $X$ is a random variable with values of a measure space $(A,\mathcal A,\mu)$ and with pdf $f:A\to [0,1]$, then for all measurable set $S\in\mathcal A$, $$P(X\in S) = \int_S fd\mu $$ So, if $A=\Bbb R$ (and $\mu=\lambda$), then $$P(a<X<b)=\int_a^b f(x)dx$$ So, $f(x) = \displaystyle\lim_{t\to 0} \frac1{2t}\int_{x-t}^{x+t} f =\lim_{t\to 0} \frac1{2t} P(|X-x|<t) $ for example.. We can call it 'relative likelihood'..

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This is a better answer than Alex's but doesn't explain the significance of the number $f(x)$. Does it have a meaning independent of a cdf? Andre's answer of it being approximately $hf(a)$ is great but he doesn't indicate if there's more to $f(x)$ by itself. – Jacob Feb 27 '13 at 17:23

Intro statistics focuses on the PDF as the description of the population, but in fact it is the CDF (cumulative density function) that gives you a functional understanding of the population, as points on the CDF denote probabilities over a relevant range of measures. If you look at all stats from this perspective, then the PDF is just the description of probability change with respect to a change around a point along the measure at hand. The values on the PDF therefore only give you a look at the spread. For example, given two normal distributions $N(\mu_1, \sigma_1^2)$ and $N(\mu_2, \sigma_2^2)$, if you choose any value of $x$ to get point $p_n=\mu_n+x\cdot\sigma_n$ for the respective distributions and get $X_1[p_1 ] > X_2[p_2 ]$, then this just means $\sigma_1 < \sigma_2$. Similar relationships exist for other distributions.

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