Question

If a random variable X has a beta distribution, its probability density function is

fX (x) = 1 xα−1(1 − x)β−1 B(α,β)

for x between 0 and 1 inclusive. The pdf is zero outside of [0,1]. The B() in the denominator is the beta function, given by beta(a,b) in R.

Write your own version of dbeta() using the beta pdf formula given above. Call your function mydbeta(). Your function can be simpler than dbeta(): use only three arguments (x, shape1, and shape2) and don’t bother with ncp or log.

Use your function mydbeta() for this part. Experiment with different values of α and β and plot the resulting beta pdf to answer the following questions. The parameters α and β have to be greater than zero. Otherwise there’s no restriction. In R, plot 3 pdf’s for each one.

1. What relationship between α and β produces a pdf which is skewed to the right?

2. What values for α and β produce a pdf which is bell-shaped? It should look close to normal.

3. Find values for α and β which produce a pdf like a concave bowl. (Concave means curved like a frown. Curved like a smile is convex.)

Answer #1

A random variable X has the cumulative distribution function
(cdf) given by F(x) = (1 + e−x ) −1 , −∞ <
x < ∞.
(i) Find the probability density function (pdf) of X.
(ii) Roughly, take 10 points in the range of x (5 points below 0
and 5 points more than 0) and plot the pdf on these 10 points. Does
it look like the pdf is symmetric around 0?
(iii) Also, find the expected value of X.

A random variable X with a beta distribution takes on values
between 0 and 1, with unknown α and β.
a) Use the method of moments to obtain an estimator
forαandβ.
b) Are the estimators sufficient statistics?

Let X be a random variable with probability density function
fX(x) given by fX(x) = c(4 − x ^2 ) for |x| ≤ 2 and zero
otherwise.
Evaluate the constant c, and compute the cumulative distribution
function.
Let X be the random variable. Compute the following
probabilities.
a. Prob(X < 1)
b. Prob(X > 1/2)
c. Prob(X < 1|X > 1/2).

Suppose a random variable X has cumulative distribution function
(cdf) F and probability
density function (pdf) f. Consider the random variable Y =
X?b
a for a > 0 and real b.
(a) Let G(x) = P(Y x) denote the cdf of Y . What is the
relationship between the functions
G and F? Explain your answer clearly.
(b) Let g(x) denote the pdf of Y . How are the two functions f
and g related?
Note: Here, Y is...

Let X ∼ Beta(α, β).
(a) Show that EX 2 = (α + 1)α (α + β + 1)(α + β) .
(b) Use the fact that EX = α/(α + β) and your answer to the
previous part to show that Var X = αβ (α + β) 2 (α + β + 1).
(c) Suppose X is the proportion of free-throws made over the
lifetime of a randomly sampled kid, and assume that X ∼ Beta(2,
8)
....

A random variable X takes values between -2 and 4 with
probability density function (pdf)
Sketch a graph of the pdf.
Construct the cumulative density function (cdf).
Using the cdf, find )
Using the pdf, find E(X)
Using the pdf, find the variance of X
Using either the pdf or the cdf, find the median of
X

5. Consider the random variable X with the following
distribution function for a > 0, β > 0:
FX (z) = 0 for z ≤ 0
= 1 – exp [–(z/a)β] for z > 0 (where exp y = ey)
(a) Determine the inverse function of FX (z), where 0 < z
< 1.
(b) Let a = β = 2 for the random variable X, and define the
numbers u1 = .33 and u2 = .9. Use the inverse...

For each of the random quantities X,Y, and Z, defined below
(a) Plot the probability mass function PMS (in the discrete
case) , or the probability density function PDF (in the continuous
case)
(b) Calculate and plot the cumulative distribution function
CDF
(c) Calculate the mean and variance, and the moment function
m(n), and plot the latter.
The random quantities are as follows:
X is a discrete r.q. taking values k=0,1,2,3,... with probabilities
p(1-p)^k, where p is a parameter with...

Let X be a random variable with probability density function fX
(x) = I (0, 1) (x). Determine the probability density function of Y
= 3X + 1 and the density function of probability of Z = - log
(X).

Let X be a random variable with the probability density function
fx(x) given by:
fx(x)=
1/4(2-x), 0<x<2
1/4(x-2), 2<=x<4
0, otherwise.
Let Y=|X-3|. Compute the probability density function of Y.

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