If a continuous random process has a probability density
function f(x) = a + bx, for...
If a continuous random process has a probability density
function f(x) = a + bx, for 0 < x < 5, where a and b are
constants, and P(X > 3) = 0.3 Determine:
If a continuous random process has a probability density
function f(x) = a + bx, for 0 < x < 5, where a and b are
constants, and P(X > 3) = 0.3 Determine:
The values of a and b.
The cumulative distribution function F(x)
P(X < 2)
P(2 < X < 4)
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 −...
A continuous probability density function (PDF) f ( X )
describes the distribution of continuous random variable X .
Explain in words, and words only, this property: P ( x a < X
< x b ) = P ( x a ≤ X ≤ x b )
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 )
=
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
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.
Consider a continuous random vector (Y, X) with joint
probability density function
f(x, y) = 1
for 0 < x < 1, x < y < x + 1.
What is the marginal density of X and Y? Use this to compute
Var(X) and Var(Y)
Compute the expectation E[XY]
Use the previous results to compute the correlation Corr (Y,
X)
Compute the third moment of Y, i.e., E[Y3]
Consider a continuous random vector (Y, X) with joint
probability density function f(x, y) = 1 for 0 < x < 1, x
< y < x + 1.
A. What is the marginal density of X and Y ? Use this to compute
Var(X) and Var(Y).
B. Compute the expectation E[XY]
C. Use the previous results to compute the correlation Corr(Y,
X).
D. Compute the third moment of Y , i.e., E[Y3].
The probability density function of the continuous random
variable X is given by
fX (x) = kx, (0 <= x <2)
= k (4-x), (2 <= x <4)
= 0, (otherwise)
1) Find the value of k
2)Find the mean of m
3)Find the Dispersion σ²
4)Find the value of Cumulative distribution function FX(x)
Let X be a continuous random variable with a probability density
function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) =
e −x
a. Find the expression for the probability density function fY
(y).
b. Find the domain of the probability density function fY
(y).