Question

If X is a continuous random variable with pdf f(x) on the interval [a,b] then show that a<E(X)<b.

Answer #1

**Answer:**

**Given that:**

If X is a continuous random variable with pdf f(x) on the interval [a,b] then show that a<E(X)<b.

From the Properties of expectaion of a random variable, we know that if X & Y be two random variables follows the same pdf or distribution. Then,

(I) if X >Y ⇒ E(X) > E(Y) and

(ii) if X < Y ⇒ E(X) < E(Y)

Let us consider X> a ⇒ E(X) > E(a)=a---------(1)

(since, expection of a constant is constant)

Let us consider X < b ⇒ E(X) < E(b)=b---------(2)

From equation (1) & (2), one can say that a < E(X) < b

Let
X be a continuous random variable rv distributed via the pdf f(x)
=4e^(-4x) on the interval [0, infinity].
a) compute the cdf of X
b) compute E(X)
c) compute E(-2X)
d) compute E(X^2)

Suppose that X is continuous random variable with PDF f(x) and
CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly
infinite) interval of the real numbers then F(x) is a strictly
increasing function of x over that interval. [Hint: Try proof by
contradiction]. (b) Under the conditions described in part (a),
find and identify the distribution of Y = F(x).

a continuous random variable X has a pdf f(x) = cx, for
1<x<4, and zero otherwise.
a. find c
b. find F(x)

2. Let X be a continuous random variable with pdf given by f(x)
= k 6x − x 2 − 8 2 ≤ x ≤ 4; 0 otherwise.
(a) Find k.
(b) Find P(2.4 < X < 3.1).
(c) Determine the cumulative distribution function.
(d) Find the expected value of X.
(e) Find the variance of X

Let X and Y be continuous random variable with joint pdf
f(x,y) = y/144 if 0 < 4x < y < 12 and
0 otherwise
Find Cov (X,Y).

A
continuous random variable X has a pdf of the form: f(x)=(951/377)
x^3, for 0,46<X<1,13. Calculate the standard deviation
(sigma) of X.

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Let X be a random variable with pdf f(x)=12,
0<x<2.
a) Find the cdf F(x).
b) Find the mean of X.
c) Find the variance of X.
d) Find F (1.4).
e) Find P(12<X<1).
f) Find PX>3.

Consider continuous random variables X and Y whose joint pdf is
f(x, y) = 1 with 0 < y < 1 − |x|. Show that Cov(X, Y ) = 0
even though X and Y are dependent. Note: For this problem, you only
need to show that the covariance is zero. You need not show that X
and Y are dependent.

A continuous random variable X has pdf ?x(?) = (? + 1) ?^2, 0 ≤
? ≤ ? + 1, Where B is the
last digit of your registration number (e.g. for FA18-BEE-123,
B=3).
a) Find the value of a
b) Find cumulative distribution function (CDF) of X i.e. ??
(?).
c) Find the mean of X
d) Find variance of X.

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