- 7 Problem 4 15 Points Let X1 X2 Xn Be I I D Random Variables With The Common Pdf 0 For X 0 F X For 0 1 (97.14 KiB) Viewed 35 times
Let
X1,X2,…,
Xn be i.i.d. random variables with the
common pdf f(x)=0, for x<0,
1/a, for 0≤x≤a, 0, for a<x,
where a>0 is a constant but its value is unknown to
you. Answer the following questions.
(a) When a random variable Y has the pdf
f_Y(y)= δ(y-a), find its
CDF and sketch the graph of the CDF.
(b) Derive the CDF of a random varialbe
Z≜max(X1,X2,…,
Xn) as a function of z, and sketch the
graph of the CDF
(c) When n is sufficiently large and the event
{X1=x1,
X2=x2, …,
Xn=xn} occurs, you
are going to infer the value of a as a
≈max(x1, x2,
…, xn(. Justify this inference method.
(Hint. Use the results in (a)and
(b).)
7 Problem 4. (15 points) Let X1, X2, ... , Xn be i.i.d. random variables with the common pdf 0, for x < 0, f(x) = for 0 < x <a, 0, for a < x, where a > 0 is a constant but its value is unknown to you. Answer the following questions. 1 = 7 a = — (a) (5 points) When a random variable Y has the pdf fy(y) = 8(y – a), find its CDF and sketch the graph of the CDF. (b) (5 points) Derive the CDF of a random variable Z 4 max(X1, X2, ... , Xn) . as a function of 2, and sketch the graph of the CDF. (c) (5 points) When n is sufficiently large and the event {X1 = x1, X2 = 22, ... , Xn Xn} occurs, you are going to infer the value of a as = a = max(x1, X2, ... ,xn). Justify this inference method. (Hint. Use the results in (a) and (b).)