k-armed bandit problem in Python
Python is notoriously slow with native loops; i.e.

for j in range(nRun):
    ...
    for t in range(T):
        ...


is probably your bottleneck. I haven't tried to understand necessarily what you are coding but can you vectorise some calculations?

Something like 

sigma = np.ones(nActions)
mu = np.array([-0.4677,-0.1249,1.4790,-0.8608,0.7847,0.3086,-0.2339,-1.0570,-0.2841,-0.0867])
def Reward(action:int, mu:np.ndarray, sigma:np.ndarray):
    return mu[action] + sigma[action]*np.random.normal()


is better replaced by:

temp_rvs = np.random.rand(nRun * T, 10)
reward_arr = mu + sigma * temp_rvs


then you can just index an existing variable instead of generating random variables each time through a function in your loop, i.e. you can:

reward[(j+1)*T, a]


But this takes up \$10 * nRun * T * 8\$ bytes of RAM to store the random variables.

I answered a similar question to this here: How to simulate this Gamma expansion in a Python way perhaps it might give you some ideas.