1 xe Let f(x)={? x 8. Prove that f(x) continuous only at +1. Let f(x)= $3.x...
1-> X- Let f :S → R and g:S → R be functions and c be a cluster point. Assume lim f (x), lim g(x) exists. Using the definition of the limit prove the following lim(af (x) + Bg(x)) =a lim f(x) + Blim g(x) for any a,ßeR xc XC X-> b. lim( f(x))} = (lim f(x)) f(x) lim f (x) c. If (Vxe S)g(x) # () and lim g(x)() then prove lim X-C XC 10 g(x) lim g(x) X-C
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and E[X].
[10] 1. Solve the first-order equation Tư + (x – 2) = -xe-3, > 0.
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
Let X be a continuous random variable with density f(x) = e?5x, x > b. Find b. A. (ln 5)=5 B. ?(ln 5)=5 C. e?5=5 quad D. 3 E. None of the preceding Let X be a continuous random variable with density f(x) = e-5x, x > b. Find b. A. (In 5)/5 B. — (In 5)/5 C. e-5/5 quad D. 3 E. none of the preceding
Let X and Y be continuous random variables with joint distribution function: f(x,y) = { ** 0 <y < x <1 otherwise What is the P(X+Y < 1)?
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
1. Let x, a € R. Prove that if a <a, then -a < x <a.