Question
In: Statistics and Probability
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,....
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
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
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
: Let X denote the result of a random experiment with the following cumulative distribution function...
: Let X denote the result of a random experiment with the
following cumulative distribution function (cdf): 0, x <1.5 | 1/
6 , 1.5<=x < 2 | 1/ 2, 2 <= x <5 | 1 ,x >= 5
Calculate ?(1 ? ≤ 6) and ?(2 ≤ ? < 4.5)
b. Find the probability mass function (pmf) of ?
d. If it is known that the result of the experiment is integer,
what is the probability that the result is...
Explain briefly how the cumulative distribution function of a random variable can be used to calculate...
Explain briefly how the cumulative distribution function of a
random variable can be used to calculate probabilities.
Let T > 0 be a continuous random variable with cumulative hazard function H(·). Show that...
Let T > 0 be a continuous random variable with cumulative
hazard function H(·).
Show that H(T)∼Exp(1),where Exp(λ) in general denotes the
exponential distribution with rate parameter λ.
Hints:
(a) You can use the following fact without proof: For any
continuous random variable T with cumulative distribution function
F(·), F(T) ∼ Unif(0,1). Hence S(T) ∼ Unif(0,1).
(b) Use the relationship between S(t) and H(t) to derive that
Pr(H(T) > x)= e−x.
A probability distribution function P(x) for a random variable X is defined by P(x) = P...
A probability distribution function P(x) for a random variable X
is defined by P(x) = P r{X ≤ x}. Suppose that we draw a list of n
random variables X1, X2, X3 · · · Xn from a continuous probability
distribution function P that is computable in O(1) time. Give an
algorithm that sorts these numbers in linear average case time.
Considering X a continuous random variable defined by the distribution function f(x) = 0 if x<1...
Considering X a continuous random variable defined by the
distribution function
f(x) = 0 if x<1
-k +k/x if 1<= x < 2
1 if 2<x
Find k.
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,...
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