Reference no: EM132397432

3.B. (4431) Assume that U1, U2, . . . is a sequence of i.i.d. r.v.'s, each uniformly distributed over the interval (0, 1). (I.e., P(U_i ≤ t) = t for every t between 0 and

1.) For each n = 1, 2, . . ., define the random variable M_n = min{U₁, U₂, . . . , U_n}.

(i) Show that the distribution function of M_n is
P(M_n ≤ t) = 1 - (1 - t)ⁿ   for t ∈ (0, 1).

Use this to calculate the probability density function and the expected value of M_n.

(ii) Using the distribution function explicitly, show that M_n/E(M_n) converges in distribution to a random variable having the exponential distribution with parameter 1.

3.C. (6604) This generalizes Problem 3.B. Assume that X₁, X₂, . . . is an i.i.d. sequence of nonnegative r.v.'s, and that X_i has a probability density function f(x) that is continuous on [0, c) for some positive number c (perhaps c = +∞). For each n = 1, 2, . . ., define the random variable

Mn = min{X₁, X₂, . . . , X_n}.

(i) Assume that 0 < f(0) < +∞. Find an expression for the distribution function of Mn analogous to the expression in 3.B(i) above. Use this to show that nMn converges in distribution to an exponentially distributed random variable.

(ii) Assume now that f(x) = xe^-x for x ≥ 0. In this case, the argument of part (i) would show that nM_n does not converges in distribution to a finite random variable. Show however that that √nM_n converges in distribution to a (finite) random variable that is non-constant and not exponentially distributed.

3.D. Assume that the random variable W has an exponential distribution with parameter β. Also assume that given W, the distribution of Y is Poisson with parameter W. That is,
P( Y = j | W = w ) = e ^-w(w^j / j!) ,           j = 0, 1, . . .

(i) Show that the moment generating function of Y is

MY (t) = (β/ (β - e^t + 1)).

(Hint: Condition on W.)

(ii) Show that E(Y ) = 1/β. Use the result of (i) to calculate Var(Y ).

(iii) Compute the limit of the MGF of βY (i.e., Y /E(Y )) in the limit as β → 0.

The limit is the MGF of what distribution?

(iv) Interpret the result of (iii) as a statement about convergence in distribution. Use this to obtain an approximation of P(Y≤ 150) when β = 0.01.

3.E. (6604) Assume that X_n converges in distribution to X, and that Y_n converges in distribution to Y , and that all of these random variables are defined on the same probability space. (We assuming nothing about independence.)

(i) Show by example that it is not necessarily true that X_n + Y_n converges in distribution to X + Y .

(ii) Assume that Y is constant, i.e. there exists a number c such that P(Y = c) = 1.

Prove that X_n + Y_n converges in distribution to X + Y .