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An identity in [[Sequential analysis|sequential analysis]] which states that the mathematical expectation of the sum  $  S _  \tau  = X _ {1} + \dots + X _  \tau  $
of a random number  $  \tau $
of independent, identically-distributed random variables  $  X _ {1} , X _ {2} \dots $
is equal to the product of the mathematical expectations  $   {\mathsf E}  X _ {1} $
and  $   {\mathsf E}  \tau $:

$$ 
{\mathsf E} ( X _ {1} + \dots + X _  \tau  )  = \ 
{\mathsf E} X _ {1} \cdot {\mathsf E} \tau .
$$

A sufficient condition for the Wald identity to be valid is that the mathematical expectations  $  {\mathsf E} | X _ {1} | $
and  $  {\mathsf E} \tau $
in fact exist, and for the random variable  $  \tau $
to be a Markov time (i.e. for any  $  n = 1, 2 \dots $
the event  $  \{ \tau = n \} $
is determined by the values of the random variables  $  X _ {1} \dots X _ {n} $
or, which is the same thing, the event  $  \{ \tau = n \} $
belongs to the  $  \sigma $-
algebra generated by the random variables  $  X _ {1} \dots X _ {n} $). 
Wald's identity is a particular case of a fundamental theorem in sequential analysis stating that

$$ \tag{* }
{\mathsf E} \left [ e ^ {\lambda S _  \tau  } ( \phi ( \lambda )) ^ {- \tau }
\right ]  =  1
$$

for all complex  $  \lambda $
for which  $  \phi ( \lambda ) = {\mathsf E} e ^ {\lambda X _ {1} } $
exists and  $  | \phi ( \lambda ) | \geq  1 $. 
It was established by A. Wald [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Wald, "Sequential analysis", Wiley (1952)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1957) pp. Chapt.14</TD></TR></table>

====Comments====
The general result (*) is (also) referred to as Wald's formula.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. [A.V. Skorokhod] Skorohod,   "Random processes with independent increments" , Kluwer  (1991)  pp. 23  (Translated from Russian)</TD></TR></table>