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 −...
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)
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 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).
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.
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)
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)
The joint probability density function for two continuous random
variables ?? and ?? is
??(??, ??) = ??(3??2 − ??), 0 < ?? < 1, 0 < ?? <
1.
Answer the following:
(a) Find the value of ?? such that ??(??, ??) is a valid
probability density function.
(b) Find the marginal probability density function for ??,
??(??).
(c) Find the marginal probability density function for ??,
??(??).
(d) Find the conditional probability density function for ??|?? =
??,...
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 )
A random variable has a triangular probability density function
with a = 50, b = 375, and m = 250.
What is the probability that the random variable will assume a
value between 70 and 250? If required, round your answer to four
decimal places.
What is the probability that the random variable will assume a
value greater than 280? If required, round your answer to four
decimal places.