## pr.probability – maximum of the sums of iid \$ X_i \$ s, where \$ X_i \$ is the difference between two exponential values ​​of r.v.

given $$X_i = A_i – B_i$$ from where $$A_i sim text {Exp} ( alpha)$$ and $$B_i sim text {Exp} ( lambda)$$, Define $$S_k = sum_ {i = 1} ^ k X_i$$ With $$S_0 = 0$$, and
$$M_n = max_ {1 leq k leq n} S_k.$$
Can you calculate the amount? $$mathbb {P} (M_n leq x)$$ expressly? I tried it and the result is down, but it is not clear …

In Feller's Introduction to Probability Theory and its application, he has repeatedly remarked that this type of two-exponential divergence distribution is a rare but important case in which almost all erroneous calculations can be made explicit. (V1.8 example (b) page 193; XII.2 example (b) page 395; XII.3 example (b) page 401) Unfortunately, I could not find any detailed calculations in the book.

A second reference that I have looked at is the paper "On the distribution of the maximum of sums of independent and equally distributed random variables"by Lajos Takacs (Adv. Appl. Prob 1970) Takacs mentioned that we can calculate in some special cases $$mathbb {P} (M_n leq x)$$ light. After his example on page 346 (where he only suspected) $$X_i = A_i – B_i$$ from where $$B_i$$ is exponential and $$A_i$$ is not negative) that I could count on $$A_i sim text {Exp} ( alpha)$$, $$B_i sim text {Exp} ( lambda)$$, we have
$$U (s, p) = sum_ {n = 0} ^ infty mathbb {E} left[e^{-sM_n}right]p ^ n = frac { lambda – frac {s lambda} { gamma (p)}} { lambda – s – frac { lambda alpha p} { alpha + s}}$$
from where $$gamma (p) = frac { lambda – alpha + sqrt {( alpha + lambda) ^ 2 – 4 alpha lambda p}} {2}$$a zero of the denominator above. Is there a way to simplify this to get an explicit formula for $$mathbb {P} (M_n leq x)$$?

## Random Variables – Expected value of a r.v. in terms of his mode

To let $$X$$ a random variable having an exponential distribution such that

$$E[X]= frac {1} {10}$$

Which is the probability that $$X$$ is smaller than the mode $$nu$$?

My intuition is that this probability is the same $$0$$ since $$frac {1} {10}$$ is in the first quadrant and the function goes on $$+ infty$$ However, I am not sure if I interpreted the question well.