Let X be a exponential random variable with pdf f(x) = λe−λx for
x > 0, and cumulative distribution function F(x).
(a) Show that F(x) = 1−e −λx for x > 0, and show that this
function satisfies the requirements of a cdf (state what these are,
and show that they are met). [4 marks]
(b) Draw f(x) and F(x) in separate graphs. Define, and identify
F(x) in the graph of f(x), and vice versa. [Hint: write the
mathematical relationships,...
2. Let X be exponential with rate lambda. What is the pdf of Y =
X^0.5? How about Y = X^3? Contrast the complexity of this result to
transformation of a discrete random variable.
Let X have the pdf fX(x) = 3(1 − x) 2 , 0 < x < 1.
(a) Find the pdf of Y = (1 − X) 3 . Specify the distribution of
Y (name and parameter values). (b) Find E(Y ) and Var(Y ).
Let X and Y be independent Exponential random variables with
common mean 1.
Their joint pdf is f(x,y) = exp (-x-y) for x > 0 and y > 0
, f(x, y ) = 0 otherwise. (See "Independence" on page 349)
Let U = min(X, Y) and V = max (X, Y).
The joint pdf of U and V is f(u, v) = 2 exp (-u-v) for 0 < u
< v < infinity, f(u, v ) = 0 otherwise....
Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x
for 0<X< ∞, and 0 elsewhere. Find the density function and
cdf of
a) X^1/2
b)X=e^x
c)X=1/X
Which of the random variables-X, X^1/2, e^x, 1/X does not have a
finite mean?
Let X be a uniform random variable with pdf f(x) = λe−λx for x
> 0, and cumulative distribution function F(x).
(a) Show that F(x) = 1−e −λx for x > 0, and show that this
function satisfies the requirements of a cdf (state what these are,
and show that they are met). [4 marks]
(b) Draw f(x) and F(x) in separate graphs. Define, and identify
F(x) in the graph of f(x), and vice versa. [Hint: write the
mathematical relationships,...
Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤ y ≤ 1. (i)
Find the conditional means of X given Y, and Y given X. (ii) Find
the conditional variance of X given Y. (iii) Find the correlation
coefficient between X and Y.
Let C1 be the part of the exponential curve y = πe^x where 0 ≤ x
≤ 1. Let C2 be the line segment between (1, πe) and (π, 2π). If C
is the union of these two curves, oriented from left to right, find
the work done by the force field
F = <sin(x)e^ cos(x) +y 2 , 2xy−2 sin(y) cos(y)> as a
particle moves along C
2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 −
x), 0 < x < 1,
0 elsewhere.
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Find P(−0.5 < X < 0.3).
(c) Find the median of X.