I am doing my research in Functional Analysis, especially in "Generalized inverse of Linear Maps".

I have come across Probability by studying only the methods or Distributions(like Binomial, poisson, normal,etc)

Now I wish to study the mathematical background and intuitive way of looking on it.

Can you suggest some probability text book which explains,

$\bullet$ Geometrical ideas of the Probability Distributions. May be using diagrams, graphs, etc.

$\bullet$ Proofs for the distribution functions.

$\bullet$ Problems with natural solutions and then generalizations Like, for a binomial distribution(how they are giving the probability mass function).