Question
In: Statistics and Probability
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.
Solutions
orchestra answered 1 year agoRelated Solutions
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,....
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
Plot the probability mass function (PMF) and the cumulative distribution function (CDF) of 3 random variables...
Plot the probability mass function (PMF) and the cumulative
distribution function (CDF) of 3 random variables following (1)
binomial distribution [p,n], (2) a geometric distribution [p], and
(3) Poisson distribution [?]. You have to consider two sets of
parameters per distribution which can be chosen arbitrarily. The
following steps can be followed: Setp1: Establish two sets of
parameters of the distribution: For Geometric and Poisson
distributions take two values of p (p1 and p2) and take two values
of [?],...
Plot the probability mass function (PMF) and the cumulative distribution function (CDF) of 3 random variables...
Plot the probability mass function (PMF) and the cumulative
distribution function (CDF) of 3 random variables following (1)
binomial distribution [p,n], (2) a geometric distribution [p], and
(3) Poisson distribution [?]. You have to consider two sets of
parameters per distribution which can be chosen arbitrarily. The
following steps can be followed
Plot the probability mass function (PMF) and the cumulative distribution function (CDF) of 3 random variables...
Plot the probability mass function (PMF) and the cumulative
distribution function (CDF) of 3 random variables following (1)
binomial distribution [p,n], (2) a geometric distribution [p], and
(3) Poisson distribution [?]. You have to consider two sets of
parameters per distribution which can be chosen arbitrarily. The
following steps can be followed: Setp1: Establish two sets of
parameters of the distribution: For Geometric and Poisson
distributions take two values of p (p1 and p2) and take two values
of [?],...
: 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...
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.
Briefly explain how the shape of the BE distribution function solves the UV catastrophe.
Briefly explain how the shape of the BE distribution function
solves the UV catastrophe.
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