In each time period a certain stock either goes down 1 with

In each time period, a certain stock either goes down 1 with probability .39, remains the same with probability .20, or goes up 1 with probability .41. Asuming that the changes in successive time pe- riods are independent, approximate the probability that, after 700 time periods, the stock will be up more than 10 from where it started.

Solution

Consider the following table:

Thus, for each step

mean = 0.02
variance = E(x^2) - E(x)^2 = 0.8 - 0.02^2 = 0.7996

Hence,

standard deviation = 0.894203556

Saying it went up more than 10 after 700 periods is like saying \"probability that the mean per step is greater than 10/700 = 0.014285714\".

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    0.014285714      
u = mean =    0.02      
n = sample size =    700      
s = standard deviation =    0.894203556      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -0.169073133      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.169073133   ) =    0.567130441 [ANSWER]