Question

Problem #3. X is a random variable with an exponential
distribution with rate λ = 7 Thus the pdf of X is f(x) = λ
e^{−λx}

for 0 ≤ x where λ = 7.

PLEASE ANSWER these parts if you can.

**f) Calculate the probability that X is at least .3 more
than its expected value.Use the pexp function:**

**g) Copy your R script for the above into the text box
here.**

Answer #1

TOPIC:Exponential distribution and probability.

**Required R code and output:**

Problem #3. X is a random variable with an exponential
distribution with rate λ = 7 Thus the pdf of X is f(x) = λ
e−λx for 0 ≤ x where λ = 7.
a) Using the f(x) above and the R integrate function calculate the
expected value of X.
b) Using the f(x) above and the R integrate function calculate the
expected value of X2
c) Using the dexp function and the R integrate command calculate
the expected value...

If X is an exponential random variable with parameter λ,
calculate the cumulative distribution function and the probability
density function of exp(X).

Suppose that X|λ is an exponential random variable with
parameter λ and that λ|p is geometric with parameter p. Further
suppose that p is uniform between zero and one. Determine the pdf
for the random variable X and compute E(X).

Given the exponential distribution f(x) = λe^(−λx), where λ >
0 is a parameter. Derive the moment generating function M(t).
Further, from this mgf, find expressions for E(X) and V ar(X).

A random variable XX with distribution
Exponential(λ)Exponential(λ) has the memory-less
property, i.e.,
P(X>r+t|X>r)=P(X>t) for all r≥0 and
t≥0.P(X>r+t|X>r)=P(X>t) for all r≥0 and t≥0.
A postal clerk spends with his or her customer has an
exponential distribution with a mean of 3 min3 min. Suppose a
customer has spent 2.5 min2.5 min with a postal clerk. What is the
probability that he or she will spend at least an additional 2 min2
min with the postal clerk?

Question 7) Suppose X is a Normal random variable with with
expected value 31 and standard deviation 3.11. We take a random
sample of size n from the distribution of X. Let X be the sample
mean. Use R to determine the following:
a) Find the probability P(X>32.1):
b) Find the probability P(X >32.1) when n = 4:
c) Find the probability P(X >32.1) when n = 25:
d) What is the probability P(31.8 <X <32.5) when n =
25?...

Question 6) Suppose X is a random variable taking on possible
values 0,2,4 with respective probabilities .5, .3, and .2. Y is a
random variable independent from X taking on possible values 1,3,5
with respective probabilities .2, .2, and .6. Use R to determine
the following.
f) Find the expected value of X*Y. (i.e. X times Y)
g) Find the expected value of 3X - 5Y.
h) Find the variance of 3X - 5Y
i) Find the expected value of...

Use Rstudio to compare the cdf and pdf of an exponential random
variable with rate λ=2λ=2 with the cdf and pdf of an exponential
random variable with rate 1/2.

Let X be a random variable with probability density function
f(x) = { λe^(−λx) 0 ≤ x < ∞
0 otherwise } for some λ > 0.
a. Compute the cumulative distribution function F(x), where F(x)
= Prob(X < x) viewed as a function of x.
b. The α-percentile of a random variable is the number mα such
that F(mα) = α, where α ∈ (0, 1). Compute the α-percentile of the
random variable X. The value of mα will...

Let X be an exponential random variable with parameter λ > 0.
Find the probabilities P( X > 2/ λ ) and P(| X − 1 /λ | < 2/
λ) .

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