• (a + b) n = Pn k=0 n k a k b n−k , where a, b > 0;

Probability Calculus 2019/2020 Introductory Problem Set

1. Check that:

• ^{ n} _k ^ = ^ _n−k ^{n } and ^{ n} _k ^ + ^ _k+1 ^{n } = ^{ n+1} _k+1 ^ for 0 6 k 6 n;

• ^{ n} _k ^ = ⁿ _k ^{ n−1} _k−1 ^ 0 6 k 6 n;

• (a + b) ⁿ = ^{P n} _k=0 ^{ n} _k ^ a ^k b ^n−k , where a, b > 0;

• ^{P ∞} _k=0 x ^k = _1−x ¹ oraz ^{P ∞} _k=1 kx ^k−1 = _(1−x) ¹

.

2. Calculate:

• ^{P n} _k=1 k ^{ n} _k ^ a ^k b ^n−k , where a, b > 0;

• ^{P ∞} _k=0 ^λ

_k! ^x

and ^{P ∞} _k=0 k ^λ

_k! ^x

3. Find

• ^R ₀ ^∞ exp(−ax)dx, where a > 0;

• ^R ₀ ^∞ xe ^−ax dx, where a > 0;

• ^R ₀ ¹ x ^p dx where p > −1;

• ^R ₁ ^∞ x ^−p dx, where p > 1;

• ^R ₀ ^∞ _1+x ¹

dx.

• ^R ₀ ^π sin xdx.

4. Let A, B, C be events. Using the notation with operations on sets, how would you write

“exactly two among events A, B and C occurred”?

5. Explain what we mean by

• variations with repetitions

• variations without repetitions

• permutations

• combinations