Understanding Probability Density Functions and Their Properties

[Leo Liu]

Homework Statement

N/A

Relevant Equations

##\int f(x) \cdot{x} \mathrm{d} x##

My questions are as follows:
1. How do we find them and why do we need them?
2. What are the meanings of the mean and the median of a PDF? Are the formulae below correct?
$$\int_{a}^{median} f(x) \mathrm{d}x = \int_{median}^{b} f(x) \mathrm{d}x$$
$$\int_{a}^{mean} f(x) \cdot x \mathrm{d}x = \int_{mean}^{b} f(x) \cdot x \mathrm{d}x = \frac 1 2$$

Thank you.

 

[etotheipi]

The idea is that the probability density function of a continuous random variable ##X##, ##f_X(x)##, is a probability per unit increment of ##x##. If ##f_X(x)## is constant within a certain interval, then the probability of the result being in that interval is just the probability per unit increment multiplied by the width of the interval, ##P = f_X(x) \Delta x##

If ##f_X(x)## is now a continuously varying function, you can imagine making ##\Delta x## small and summing lots of incremental probabilities in order to get the total probability of the result being within a certain interval $$P(x_1 \leq X \leq x_2) = \sum_{i=1}^n f_X(x_i) \Delta x$$ where ##x_i = x_1 + (i-1)\Delta x## as well as the usual definition of ##\Delta x = \frac{b-a}{n}## for ##n## strips for a Riemann sum. If you make the increments very small, i.e. ##\Delta x \rightarrow dx##, this becomes an integral $$P(x_1 \leq X \leq x_2) = \int_{x_1}^{x_2} f_X(x) dx$$ Now for your question. The median of a continuous random variable is where the cumulative probability up to that value of ##x## is 0.5, so if ##X## takes values in ##[a,b]##, $$\int_a^{\text{median}} f_X(x) dx = \int_{\text{median}}^{b} f_X(x) dx = 0.5$$ The expectation (similar to mean) of a continuous random variable ##X## also follows from the discrete case, which is $$E(X) = \sum_i p_i x_i$$ which translates into the continuous arena as $$E(X) = \int_a^b xf_X(x) dx$$ where ##[a,b]## is the whole range of possible values the c.r.v. can take.

Expectations are evaluated over the whole range of possible values of ##X##, and the cumulative probability up to the expectation is not (necessarily) 50%. So the second line of formulae you quote are incorrect.

 

[member 587159]

The idea is that the probability density function of a continuous random variable ##X##, ##f_X(x)##, is a probability per unit increment of ##x##. If ##f_X(x)## is constant within a certain interval, then the probability of the result being in that interval is just the probability per unit increment multiplied by the width of the interval, ##P = f_X(x) \Delta x##

If ##f_X(x)## is now a continuously varying function, you can imagine making ##\Delta x## small and summing lots of incremental probabilities in order to get the total probability of the result being within a certain interval $$P(a \leq X \leq b) = \sum_{i=1}^n f_X(x_i) \Delta x$$ where ##x_i = a + (i-1)\Delta x## as well as the usual definition of ##\Delta x = \frac{b-a}{n}## for ##n## strips for a Riemann sum. If you make the increments very small, i.e. ##\Delta x \rightarrow dx##, this becomes an integral $$P(a \leq X \leq b) = \int_{a}^b f_X(x) dx$$ Now for your question. The median of a continuous random variable is where the cumulative probability up to that value of ##x## is 0.5, so if ##X## takes values in ##[a,b]##, $$\int_a^{\text{median}} f_X(x) dx = \int_{\text{median}}^{b} f_X(x) dx = 0.5$$ The expectation (similar to mean) of a continuous random variable ##X## also follows from the discrete case, which is $$E(X) = \sum_i p_i x_i$$ which translates into the continuous arena as $$E(X) = \int_a^b xf_X(x) dx$$ where ##[a,b]## is the whole range of possible values the c.r.v. can take.

Expectations are evaluated over the whole range of possible values of ##X##, and the cumulative probability up to the expectation is not (necessarily) 50%. So the second line of formulae you quote are incorrect.

I think this is good intuition, but the context in which this works is larger (for example ##f_X## must not be continuous, but merely measurable).

 

[etotheipi]

I think this is good intuition, but the context in which this works is larger (for example ##f_X## must not be continuous, but merely measurable).

I admit I know little about how such objects are formally defined, my understanding on this topic is limited to quite an operational approach. In my (limited) experience, thinking of ##f_X(x) dx## as a probability and relating this to the discrete case helps me to get a hold on what's going on.

Apologies if I butchered the maths!

 

[Leo Liu]

Aha, thank you for your detailed answer!

The idea is that the probability density function of a continuous random variable ##X##, ##f_X(x)##, is a probability per unit increment of ##x##. If ##f_X(x)## is constant within a certain interval, then the probability of the result being in that interval is just the probability per unit increment multiplied by the width of the interval, ##P = f_X(x) \Delta x##

I would like to know if this explains why PDF is the derivative of the CDF because if ##\Delta P = f_X(x) \Delta x##, then it can be shown that ##\lim_{\Delta x \to 0} \frac {\Delta P(x)} {\Delta x}##.
Also, if we already knew the CDF, why would one want to find its PDF since we can calculate the probability by subtracting ##y_1## from ##y_2##?

The expectation (similar to mean) of a continuous random variable X also follows from the discrete case

I see - it is just expectation. Would you minding telling me why mathematicians don't just use expectation in this case?

 

[etotheipi]

I would like to know if this explains why PDF is the derivative of the CDF because if ##\Delta P = f_X(x) \Delta x##, then it can be shown that ##\lim_{\Delta x \to 0} \frac {\Delta P(x)} {\Delta x}##.

The CDF of a c.r.v. which takes values in the interval ##[a,b]## is defined as $$F_X(x) = \int_a^x f_X(x') dx'$$ If we take the derivative of this function w.r.t. ##x##, the fundamental theorem of calculus gives us
$$\frac{dF_X(x)}{dx} = \frac{d}{dx} \int_a^x f_X(x') dx' = f_X(x)$$ which is the desired result. Your intuition is correct, since the PDF is really just the rate of change of the cumulative probability w.r.t ##x##.

Also, if we already knew the CDF, why would one want to find its PDF since we can calculate the probability by subtracting ##y_1## from ##y_2##?

For finding probabilities, yes, it's sufficient to have the CDF. But there is a lot more you can do with c.r.v.'s, a lot of which is formulated in terms of the PDF. For instance, to find the expectation or variance, you need to use the PDF in the various integrals.

Being able to switch between them is also important. If you have a c.r.v. e.g. ##X##, and want to find the distribution of ##Z = X^2##, a common approach is to go via the CDF. In this case, (for simplicity, let's suppose ##X## takes only positive values): $$F_Z(z) = P(Z<z) = P(X^2 < z) = P(X < \sqrt{z}) = F_X(\sqrt{z})$$ and then you can differentiate w.r.t. ##z## to find the PDF of ##Z##.

I see - it is just expectation. Would you minding telling me why mathematicians don't just use expectation in this case?

I'm not sure what you mean by this. The easiest way to think about it is that for a given set of numerical data (i.e. you have already taken a sample from the distribution) you can calculate a mean. Whilst if you haven't got any actual measurements yet you are instead calculating the expectation of the c.r.v. The two concepts are similar, but quite distinct. For a probability distribution, it is the expectation that we usually talk about in this context.

 

[archaic]

Two articles that may interest you:
https://mathinsight.org/probability_distribution_idea
https://mathinsight.org/probability_density_function_idea

 

[Leo Liu]

Two articles that may interest you:
https://mathinsight.org/probability_distribution_idea
https://mathinsight.org/probability_density_function_idea

Wow, thank you for telling me this amazing website. I wish there is one for physics too.

 

[WWGD]

Re expectation, it is considered a measure of center, to understand where the values tend to aggregate*. In symmetric distributions such as the normal , mean and median ( and mode) coincide. If /when they don't, the distribution is skewed.

*Mean may not be reflective of center if there are outliers. If this last is the case, the median is used.