I am trying to understand a paper on options titled "OPTION PRICING: A SIMPLIFIED APPROACH" [1]. In it option price is calculated as the expected payoff from the possible states of stock prices by binomial distribution approach.  

I am stuck at one step.  

What does the following sentences exactly mean?



> 
> Now, the latter bracketed expression is the complementary binomial distribution function $\Phi[a; n,p]$. The first bracketed expression can also be interpreted as a complementary binomial distribution function $\Phi[a; n, p’]$.
> 
> 
> 


Thank you in advance.


**Bibliography** 


[1] John C. COX and Stephen A. ROSS and Mark RUBINSTEIN "[Option Pricing: A Simplified Approach](http://static.stevereads.com/papers_to_read/option_pricing_a_simplified_approach.pdf)", Journal of Financial Economics 7 (1979) 229-263. [DOI: 10.1016/0304-405X(79)90015-1](https://doi.org/10.1016/0304-405X(79)90015-1)



What is a complementary binomial distribution function $\Phi[a; n,p]$?
The complementary binomial distribution is defined by
$$\Phi(a;n,p)=\sum \_{j=a}^n \binom{n}{j} p^j(1-p)^{n-j},\;\;0<p<1,\;\;0\leq a\leq n.$$
see [The derivation of diffusion-jump modes for power plant projects under risk](http://www.worldresearchlibrary.org/up_proc/pdf/1124-15114160424-7.pdf). A more correct terminology would be to call $\Phi$ the *complement of the binomial cumulative distribution* function, which is how it is called in [MatLab.](https://www.mathworks.com/help/stats/binocdf.html#mw_d8519302-0c92-417f-88f9-a27b6a3f4e7f)


