Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence...
Suppose that X1,X2,... is a sequence of i.i.d. r.v.s having uniform distribution on [0,1]. Define Yn=n(1−max1≤i≤nXi) for n=1,2,.... Prove that Yn converges in distribution to an exponential distribution.
Exercise 4 Let the Xc's be i.i.d. real random variables, uniform on [0,1]. What is the limit of (X3 +..+ X3)/(X1 .+Xn) as n -> co? In which sense?
2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18 2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18
2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18 2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1): If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
Let X1,X2,...,Xn be an independent and identically distributed (i.i.d.) random sample of Beta distribution with parameters α = 2 and β = 1, i.e., with probability density function fX(x) = 2x for x ∈ (0,1). Find the probability density function of the first and last order statistics Y1 and Yn.
Let X1, X2, X3 be independent Binomial(3,p) random variables. Define Y1 = X1 + X3 and Y2 = X2 + X3. Define Z1 = 1 if Y1 = 0; and 0 otherwise. Define Z2 = 1 if Y2 = 0; and 0 otherwise. As Z1 and Z3 both contain X3, are Z1 and Z3 independent? What is the marginal PMF of Z1 and Z2 and joint PMF of (Z1, Z2) and what is the correlation coefficient between Z1 and Z2?
Problem 8. Let X1, X2, , Xn be independent ฆ(0,1) random variables. Let m,-1 for k 1,2,3. Are there numbers mi,m2, m3 such that n.y rn1 m1 a.S n3 m3 holds? If so, calculate these numbers.