Question

In: Math

Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to...

Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to be the smallest k such that X1+X2+⋯+Xn exceeds cn=n2+12n−−√, namely,

Nn = min{k≥1:X1+X2+⋯+Xk>ck}

Does the limit

limn→∞P(Nn>n)

exist? If yes, enter its numerical value. If not, enter −999.

unanswered

Submit

You have used 1 of 3 attempts Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.

Solutions

Expert Solution


Related Solutions

6. Let X1, X2, ..., X101 be 101 independent U[0,1] random variables (meaning uniformly distributed on...
6. Let X1, X2, ..., X101 be 101 independent U[0,1] random variables (meaning uniformly distributed on the unit interval). Let M be the middle value among the 101 numbers, with 50 values less than M and 50 values greater than M. (a). Find the approximate value of P( M < 0.45 ). (b). Find the approximate value of P( | M- 0.5 | < 0.001 ), the probability that M is within 0.001 of 1/2.
Let X1, X2, . . . be a sequence of independent and identically distributed random variables...
Let X1, X2, . . . be a sequence of independent and identically distributed random variables where the distribution is given by the so-called zero-truncated Poisson distribution with probability mass function; P(X = x) = λx/ (x!(eλ − 1)), x = 1, 2, 3... Let N ∼ Binomial(n, 1−e^−λ ) be another random variable that is independent of the Xi ’s. 1) Show that Y = X1 +X2 + ... + XN has a Poisson distribution with mean nλ.
2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on...
2. Let X1, X2, . . . , Xn be independent, uniformly distributed random variables on the interval [0, θ]. (a) Find the pdf of X(j) , the j th order statistic. (b) Use the result from (a) to find E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the mean difference between two successive order statistics. (d) Suppose that n = 10, and X1, . . . , X10 represents the waiting times that the n = 10...
Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1) random variables. (...
Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1) random variables. ( a) Compute the cdf of Y := min(X1, . . . , Xn). (b) Use (a) to compute the pdf of Y . (c) Find E(Y ).
Let X1 and X2 be independent UNIF(0,1) random variables and consider the transformations Y1= X1X2 and...
Let X1 and X2 be independent UNIF(0,1) random variables and consider the transformations Y1= X1X2 and Y2 =X1/X2. Find the joint pdf of Y1 and Y2 and indicate their joint support of Y1 and Y2. Show Work.
Let X, Y and Z be independent random variables, each uniformly distributed on the interval (0,1)....
Let X, Y and Z be independent random variables, each uniformly distributed on the interval (0,1). (a) Find the cumulative distribution function of X/Y. (b) Find the cumulative distribution function of XY. (c) Find the mean and variance of XY/Z.
Let X1, X2,..., Xnbe independent and identically distributed exponential random variables with parameter λ . a)...
Let X1, X2,..., Xnbe independent and identically distributed exponential random variables with parameter λ . a) Compute P{max(X1, X2,..., Xn) ≤ x} and find the pdf of Y = max(X1, X2,..., Xn). b) Compute P{min(X1, X2,..., Xn) ≤ x} and find the pdf of Z = min(X1, X2,..., Xn). c) Compute E(Y) and E(Z).
6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M =...
6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M = max(X1,...,Xn). (a) Find the density function of M. (b) Find E[M] and V[M].
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.
Let ? and ? be two independent uniform random variables such that ?∼????(0,1) and ?∼????(0,1). A)...
Let ? and ? be two independent uniform random variables such that ?∼????(0,1) and ?∼????(0,1). A) Using the convolution formula, find the pdf ??(?) of the random variable ?=?+?, and graph it. B) What is the moment generating function of ??
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT