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.

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

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.

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, 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.

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 U and V be independent continuous random variables uniformly distributed from 0 to 1. Let...

Let U and V be independent continuous random variables uniformly distributed from 0 to 1. Let X = max(U, V). What is Cov(X, U)?

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).

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 ??

Subjects