Calculus and Above

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1) A point is...

1) A point is generated at random in the plane according to the following polar scheme. A radius R is chosen, where the distribution of R^2 is X^2 with 2 degrees of freedom. Independently, an angle θ is chosen, where θ ~ uniform (0, 2π. Find the joint distribution of X=Rcosθ and Y=Rsinθ.

2) If X_n→X and Y_n→Y in probability, thenX_n+Y_n→X+Y in probability.

Let {X_i } be iid, E(X_i )=μ and Var(X_i )=σ^2. Set X ̅=(∑_(i=1)^n▒X_i )/n .

Show that 1/n ∑_(i=1)^n▒〖〖(X_i-X ̅)〗^2→σ^2 〗 in probability.

2) If X_n→X and Y_n→Y in probability, thenX_n+Y_n→X+Y in probability.

Let {X_i } be iid, E(X_i )=μ and Var(X_i )=σ^2. Set X ̅=(∑_(i=1)^n▒X_i )/n .

Show that 1/n ∑_(i=1)^n▒〖〖(X_i-X ̅)〗^2→σ^2 〗 in probability.

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