Let X be a random variable. Suppose X ∼ Exp(0.5). The area under
the pdf of...
Let X be a random variable. Suppose X ∼ Exp(0.5). The area under
the pdf of the Exp(0.5) distribution above the interval [0, 2] is
0.6321. What is the value of the cumulative distribution function
FX of X at x = 2?
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,...
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 be a continuous random variable with pdf: f(x) = ax^2 −
2ax, 0 ≤ x ≤ 2
(a) What should a be in order for this to be a legitimate
p.d.f?
(b) What is the distribution function (c.d.f.) for X?
(c) What is Pr(0 ≤ X < 1)? Pr(X > 0.5)? Pr(X > 3)?
(d) What is the 90th percentile value of this distribution?
(Note: If you do this problem correctly, you will end up with a
cubic...
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.
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0.
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Crameer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)2 or -(x/θ)^2 if you cannot read that)
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
(15 pts) Suppose that the continuous random variable X has
pdf
?(?) = {
?; 0 < ? < 2 2?; 5 < ? < 10 0; otherwise
a) Determine the value of c that makes this a legitimate pdf.
b) Sketch a graph of this pdf.
c) Determine the cumulative distribution function (cdf) of X.
d) Sketch a graph of this cdf.
e) Calculate ? = ?(?) and ? = ??(?).
f) What is ?(? = ?)?
g) Compute...
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.
Suppose that T (≥ 0) is a continuous random variable. Let its
pdf, cdf, survival function, hazard rate function, and cumulative
hazard rate function be f(t), F(t), s(t), h(t), and H(t),
respectively. Note that H(t) is defined by R t 0 h(u)du.
a. Denote s(t), h(t), and H(t) as a function of f(t).
b. Denote f(t), h(t), and H(t) as a function of s(t).
c. Denote f(t), s(t), and H(t) as a function of h(t).
d. Denote f(t) and s(t)...