Let f(x) be an integrable function such that 1 -4 -4 and 1. 2 Compute the...
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
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].
Let X be a continuous random variable with the following density function. Find E(X) and var(X). 6e -7x for x>0 f(x) = { for xso 6 E(X) = 49 var(X) =
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0 < x, y < 4 and y> 4-1, otherwise. (a) Set up but do not compute an integral to find E(XY). (b) Let fx() be the marginal probability density function of X. Set up but do not compute an integral to find fx(x) when I <r54. (c) Set up but do not compute an integral to find P(Y > X).
check if e-1/4/ f(x) if x > 0 if x < 0 is differentiable at 0.
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
Exercise 6. Show that if f(x) > 0 for all x e [a, b] and f is integrable, then Sfdx > 0.
11. Let X be a continuous random variable with density function fare-102 for 10 f(1) = lo otherwise where a > 0. What is the probability of X greater than or equal to the mode of X?