1. The random variable X has probability density function: f(x)
= ( ke−x 0 ≤ x ≤ ln 5 4 0 otherwise Part a: Determine the value of
k. Part b: Find F(x), the cumulative distribution function of X.
Part c: Find E[X]. Part d: Find the variance and standard deviation
of X. All work must be shown for this question. R-Studio should not
be used.
A random variable X has a density given by
f X ( x ) = (1 −
x^2 ) [ u ( x ) − u (
x − 1) ] + aδ ( x − 2 )where
u(x) is the unit step function and δ(x)
is a delta function.
a). Find the value of a, E(X) and σ
δx^2 .
b). Find and make a labeled sketch of
FX (x) .
C). W = the event {X ≥ 0.5}....
Assume that a continuous random variable has a following
probability density function:
f ( x ) = { 1 10 x 4 2 ≤ x ≤ 2.414 0 o t h e r w i s e
Use this information and answer questions 3a to 3g.
Question a: Which of the
following is a valid cumulative density function for the defined
region ( 2 ≤ x ≤ 2.414)?
A.F x ( x ) = 1 50 x 5 −...
1. Use the definition of convexity to prove that the function
f(x) = x2 - 4x + 8 is convex. Is
this function strictly convex?
2. Use the definition of convexity to prove that the function f(x)=
ax + b is both convex and
concave for any a•b ≠ 0.
Consider a continuous random variable X with the probability
density function f X ( x ) = x/C , 3 ≤ x ≤ 7, zero elsewhere.
Consider Y = g( X ) = 100/(x^2+1). Use cdf approach to find the cdf
of Y, FY(y). Hint: F Y ( y ) = P( Y <y ) = P( g( X ) <y )
=
The random variable X has probability density function: f(x) =
ke^(−x) 0 ≤ x ≤ ln (5/4) 0 otherwise Part a: Determine the value of
k. Part b: Find F(x), the cumulative distribution function of X.
Part c: Find E[X]. Part d: Find the variance and standard deviation
of X. All work must be shown for this question.
Let the continuous random variable X have probability density
function f(x) and cumulative distribution function F(x). Explain
the following issues using diagram (Graphs)
a) Relationship between f(x) and F(x) for a continuous
variable,
b) explaining how a uniform random variable can be used to
simulate X via the cumulative distribution function of X, or
c) explaining the effect of transformation on a discrete and/or
continuous random variable
A continuous random variable X, has the density function f(x)
=((6/5)(x^2)) , 0 ≤ x ≤ 1; (6/5) (2 − x), 1 ≤ x ≤ 2; 0, elsewhere.
(a) Verify f(x) is a valid density function. (b) Find P(X > 3 2
), P(−1 ≤ X ≤ 1). (c) Compute the cumulative distribution function
F(x) of X. (d) Compute E(3X − 1), E(X2 + 1) and σX.
The following density function describes a random variable X.
f(x)= (x/64) if 0<x<8 and f(x) = (16-x)/64 if
8<x<16
A. Find the probability that X lies between 2 and 6.
B. Find the probability that X lies between 5 and 12.
C. Find the probability that X is less than 11.
D. Find the probability that X is greater than 4.
2 Consider the probability density function (p.d.f) of a
continuous random variable X: f(x) = ( k x3 , 0 < x < 1, 0,
elsewhere, where k is a constant. (a) Find k. (b) Compute the
cumulative distribution function F(x) of X. (c) Evaluate P(0.1 <
X < 0.8). (d) Compute µX = E(X) and σX.