Given a random variable which can take on a discrete set of values x = [-0.1 -0.05, 0, 0.05, 0.1]. These values have a discrete set of probabilities p = [0.05, 0.20, 0.50, 0.20, 0.05].
Define the probability that after 2 trials of x, each evaluation is an independent event, $F(x, 2) \ge 0.1$. The function is defined as $F(x, T) = (\prod_{i=1}^{T} 1+x_i) - 1$.
I approached this problem with a multinomial distribution, but I am not sure I am including the probability vector in the correct way, and adding trials correctly.
$P(F\ge 0.1|T=1) = \frac{1}{5}\times0.05 = 0.01$.
$P(F\ge 0.1|T=2) = \frac{6}{25}\times(0.05\times0.5 + 0.05\times0.2 + 0.05\times0.05 + 0.2\times0.2 + 0.2\times0.05 +0.5\times0.05) = 0.027$
Is the total probability $\sum_{i=1}^{T}P(F\ge0.1|T_i)$?
How would you generalize this problem for say $T=5$?