What does pdf tell me that an induced probability measure doesn't? I.E what is the point of a pdf? If we use the following definition:
Let $(\Omega, \mathscr{F},P)$ be a probability space, and let $X:\Omega \to \mathbb{R}^k$ be a random vector. Let $P_X$ be the probability measure on $\mathscr{B}(\mathbb{R}^k)$induced by $X$. If $P_X$ is absolutely continuous w.r.t $k$-dimensional lebesgue measure $\mu$, with density $f_X:\mathbb{R}^k \to \mathbb{R}$, then $f_X$ is called the probability density function} of $X$, and $X$ is referred to as a continuous random variable}. For any set $B\in \mathscr{B}(\mathbb{R}^k)$, we have
$$
P_X(B) = \int_Bf_X \, d\mu = \int\dots\int1_B(x_1,\dots,x_k)f_X(x_1,\dots,x_k) \, dx_1 \dots \, dx_k
$$
Focusing on the first part, $P_X(B) = \int_Bf_X \, d\mu$, why do we WANT to know this? Does $f_X$, the pdf, tell us something that $P_X$ doesn't? I guess $P_X$ is defined on the borel sets, whereas $f_X$ is defined on $\mathbb{R}^k$... Or is the answer just something along the lines that we want to work with Lebesgue measure instead of $P_X$, and $f_X$ allows us to do so?
 A: It is somewhat unusual to specify the mapping $X:\Omega\to\mathbb R^n$. Often what one knows about $X$ is precisely the density or the cumulative distribution function.  How does one specify what the "induced probability measure" is except by specifying the density or the c.d.f.?


*

*Suppose $X,Y$ are independent random variables and $\Pr(X>t) = e^{-\alpha t}$ for $x>0$ and $\Pr(Y>t) = e^{-\beta t}$ for $t>0$, and you want to know the conditional expected value of $X$ given $X+Y$.  How do you do it?

*What if $R$ is uniformly distributed on the interval $[0,1]$ and $X_1,X_2,X_3,\ldots$ are conditionally independent given $R$, and $\Pr(X_n=1\mid R)=R$ and $\Pr(X_n=0\mid R)=1-R$, for $n=1,2,3,\ldots$, and you want the conditional distribution of $R$ given $X_1+\cdots+X_n$.  How do you find it?

*What if $\Pr(X>x) = e^{-x/\theta}$ for $x>0$ and $N\mid X\sim\mathrm{Poisson}(X)$, and you want the marginal (or "unconditional") distribution of $N$.  How do you find it?

*Say you're doing a one-way analysis of variance with i.i.d. normally distributed errors with expected value $0$.  Your test statistic for testing equality of group means is the usual F-statistic, the ratio of within-group sums of squares to between-group sums of squares each divided by respective degrees of freedom.  How do we know what the distribution of the test statistic is, given that the null hypothesis is true?

