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Find all entire functions such that $|f(z)|\leq |z^2-1|$ for all $z\in\mathbb C$.
Non-closed subspace of a Banach space

This question is from DeGroot’s “Probability and Statistics” :

Unbounded p.d.f.’s.Since a value of a p.d.f.(probability density function) is a probability density, rather than a

probability, such a value can be larger than $1$. In fact, the values of the following

p.d.f. are unbounded in the neighborhood of $x = 0$:$$f(x) =

\begin{cases}

\frac{2}{3}x^{-\frac{1}{3}} & \text{for 0<$x$<1,} \\

0 & \text{otherwise.} \\

\end{cases}$$

Now, I don’t know how the p.d.f. can take value larger than $1$.Please let me know the difference between the probability and probability density.

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Simply put:

$\rho(x) \delta x$ is the probability of measuring $X$ in $[x,x+\delta x]$.

With

$\rho(x):=$ probability density.

$\delta x:=$ interval length.

A probability will be obtained by computing the integral of $ \rho(x) $ over a given interval (i.e. the probability of getting $X\in [a,b] $ is $\int_a^b \rho(x) dx$. While $\rho(x)$ can diverge, the *integral* itself will not, and this is due to the fact that we ask that $\int_\mathbb{R}\rho(x) dx=1$, which means that the probability of measuring any outcome is 1 (we are sure that we will observe *something*). If the integral over the **whole** range gives 1, the integral over a smaller portion will give less than 1, because p.d.f. can’t be negative (a negative probability is meaningless).

The specific values $f(x)$ of the density function $f$ are the probability densities, and they express “relative probabilities”, and the main point is that for a (measurable) subset $A$ of possible values (now $A\subseteq\Bbb R$), we have

$$\int_Af\ =\ P(X\in A)$$

if the random variable $X$ has distribution described by $f$. In particular, $\int_{\Bbb R}f=1$, though its specific values, as shown by the given unlimited example, can be greater than $1$.

Probability density is a “density” FUNCTION f(X). While probability is a specific value realized over the range of [0, 1]. The density determines what the probabilities will be over a given range. What does it mean to have a probability density?

The probability density function for a given value of random variable X represents the density of probability (probability per unit random variable) at that particular value of random variable X.

Now, I don’t know how the p.d.f. can take value larger than 1

It is in this sense that probability density can take values larger than 1.

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