Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for
all
x, y ∈ R. Prove that for some constant a ∈ R, f(x) = ax.
Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all X, Y E R. Prove that for some constant a ER, f(x) = ax.
ULUM turu Problem 1. Assume that f:R R is continuous and satisfies f(x) - f(y) = (x - y)?, for all x, y ER. Show that f is constant.
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9) Prove that if the function f is continuous on a, b, then there is c E [a, b such that f(x)dax a Ja f(e)
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9)...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed
Real analysis
2. Consider the following three definitions: A function f : R-+R is lax-continuous at a E R provided for all e > 0 there is a 6 > 0 such that for all r E R, if x - al6 then |f(x)- f (a)e A function f : R - R is e-continuous at a E R provided for all e >0 there is a 6 > 0 such that for all r E R, if |a- a...
(1) If f: R₃ R a continuous function such that f(x)² > 0 for all xER. Show that either f(x) >0 for all a ER or f(x) <0 all X E R.
Assume that f is a continuous function on R such that f() = A | $(e) dt for some A > 0 and all x, and [f(x) < M for some M > 0 and all x E R. a) Prove that f has continuous derivatives of all orders on R, and for k > 1 that f(k) (x) = A(f(k-1)(x + 1) – f(k-1)(x - 1)).
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If f is differentiable on R", prove that af Экп af axi (Xi , . . . , xn) ER". for all x
Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If...
a. Suppose X and Y are continuous random variables with joint
denisty f(x,y). Prove that the density of X+Y is given by:
Use part (a) to show that if X,Y are independent and standard
Gauss-ian (i.e.N(0,1)) then X+Yi s centered Gaussian with variance
2 that is N(0,2).
fx+r(t) = { $(8,6 – u)dt