Question
In: Statistics and Probability
Given the cumulative distribution of a gamma random variable find: The probability density function Show that...
Given the cumulative distribution of a gamma random variable find:
- The probability density function
- Show that it is a valid probability function
- The moment generating function
- The Expected mean
- The variance
Solutions
orchestra answered 1 year agoRelated Solutions
Given the cumulative distribution of an exponential random variable find: The probability density function Show that...
Given the cumulative distribution of an exponential random
variable find:
The probability density function
Show that it is a valid probability function
The moment generating function
The Expected mean
The variance
Show that the cumulative distribution function for a random variable X with a geometric distribution is...
Show that the cumulative distribution function for a random
variable X with a geometric distribution is
F(x) = 0 for x < 0,
F(x) = p for 0 <= x < 1,
and, in general, F(x)= 1 - (1-p)^n for n-1 <= x < n for n
= 2,3,....
Let the continuous random variable X have probability density function f(x) and cumulative distribution function F(x)....
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
The probability density function of the continuous random variable X is given by fX (x) =...
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)
The probability density function of the random variable X is given by fX(x) = ax +...
The probability density function of the random variable X is
given by fX(x) = ax + 2/9 if 1/2 ≤ x ≤ 3, and 0 otherwise.
(a) Compute the value of a.
(b) Let the random variable Y be defined as Y = [X], where [·]
is the “round down” operator (that is, for example, [2.5] = 2,
[−2.5] = −3, [−3] = −3). Find the probability mass function of Y .
(Hint: For Y to take value k, what...
If a random variable X has a beta distribution, its probability density function is fX (x)...
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,...
If a random variable X has a beta distribution, its probability density function is fX (x)...
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,...
If a random variable X has a beta distribution, its probability density function is fX (x)...
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,...
1. Let X be a random variable with probability density function fX given by fX(x) =...
1. Let X be a random variable with probability density function
fX given by fX(x) = γαγ/ (x + α)^γ+1 , x ≥ 0,
0, x < 0,
where α > 0 and γ > 0.
(a) Find the cumulative distribution function (cdf) FX of X.
(b) Let Y = log(X+α /α) . Find the cdf of Y and identify the
distribution.
(c) How could a realisation of X be generated from an R(0,1)
random number generator?
(d) Let Z...
The random variable Y has an exponential distribution with probability density function (pdf) as follows: f(y)...
The random variable Y has an exponential distribution with
probability density function (pdf)
as follows:
f(y) = λe−λy, y >0
= 0, otherwise
(i) Showing your workings, find P (Y > s|Y > t), for s ≥
t. [3]
(ii) Derive an expression for the conditional pdf of Y ,
conditional on that Y ≤ 200. [3]
N(t) is a Poisson process with rate λ
(iii) Find an expression for the Cumulative Distribution
Function (CDF) of the waiting time until...
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