Clculate a fair bet for this modification of the game -

Question

1. Beginning

1.1 Calculate E[X] and Var(X) for a II(λ)-distributed random variable X.

Info: We know from first lesson II(λ) = ∑_k=0^∞e^-λ, λ^k/k!∈_k, λ ∈ [0, ∞)
1.2  Calculate E[X] and Var(X) for a Exp(λ)-distributed random variable X.

2 St. Petersburg Game

2.1 a

At St. Petersburg Game, a player throws a coin until he gets tail. If this happens at the n-th throw, the Player wins 2ⁿ Dollar. Let X be the random win that been paid to a player. Show that there is no fair bet in the St. Petersburg game, meaning there is no a ∈ R with E[X - a] = 0

2.2 b

We extend the St. Petersburg game with the follogin rule: If the player does not throw a tail in the first m ∈ N rounds, he wins nothing, even if in a later try he throws a tail. Calculate a fair bet for this modification of the game.

3

3.1 a

Let X a discrete, No, random variable with distribution function F. Show:

Var(X) = ∑^∞_n=0(2n + 1)(1 - F(n)) - (∑^∞_n=0(1 - F(n)))²

3.2 b

Let X a continuous, not negative random variable with distribution function F. Show:
Var(X) = ₀∫^∞ 2t(1 - F(t))dt - (₀∫^∞ 1 - F(t)dt)²

4

Let (X_n)_n∈N a sequence of independent and identical equally distribution random variable on [0, 1]

4.1 a
Calculate for all n ∈ N the distribution function F_n of Y_n := max(X₁, X_n).

4.2 b
Calculate E[Y_n] and Var(Y_n) for all n ∈ N.