Studying for a statistics exam. I have come across this problem:
and it presents to me some important and extremely basic questions (I have a LONG way to go before I'm prepared for this exam).
- Can these functions be both cdfs and pdfs/pmfs? Or are they one or the other? Is this a yes/no or a cdf/pdf/pmf question?
- What criteria do we use to evaluate if they are valid cdfs or pdfs/pmfs?
- A friend iterated I should integrate across the range of the "functiony" bit of the functions (the terms with x) and if the result is 1, that the functions are valid. But valid what?
2 Answers
$\begingroup$I think giving an answer in terms of probability axioms is not quite at the level of the OP's actual question.
Typically, we denote a density using lowercase $f$, and a cumulative distribution as uppercase $F$. A PDF must satisfy two basic criteria: (1) it must be nonnegative everywhere, and (2) it must integrate to $1$ over its support (the support is the set of $x$-values where the density is positive). A CDF must satisfy three criteria: (1) $\displaystyle \lim_{x \to -\infty} F(x) = 0$, (2) $\displaystyle \lim_{x \to +\infty} F(x) = 1$, and (3) it must be nondecreasing.
A quick sketch of the various functions will reveal if some of these conditions are violated. For (a) you need to test if $4x^4-3x^2$ is an increasing function on $[0,1]$--you would do this by calculating the derivative and seeing if it is nonnegative; for (b) you need to do the same (but it should be obvious that it is not); for (c) and (d), integrate the functions over their respective supports $[0,2]$ and $[0,1]$, and check that both functions are nonnegative in those intervals.
$\endgroup$ $\begingroup$Probability Mass functions $p_{x}(x)$or Probability Density functions $f_x(x)$ have to satisfy the 3 Probability axioms:
1. $p_x(x)\geq0$
2. $\sum_{\forall x \in \Omega} p_x(x) = 1$, where $\Omega$ is the sample space
3. Any countable sequence of disjoint events $E_1,E_2\dotsb$ satisfy $P(E_1 \bigcup E_2 \bigcup \dots )=\sum_{i=1}^{\infty}P(E_i)$
The same apply for $f_x(x)$. This is not a Yes/No question and you need to verify that the above conditions are satisfied.
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