Question

USING MATLAB:

1. Assume Y is an exponential random variable with rate parameter λ=2.

(1) Generate 1000 samples from this exponential distribution using inverse transform method

(2) Compare the histogram of your samples with the true density of Y.

Answer #1

**IF YOU HAVE ANY DOUBTS COMMENT BELOW I WILL BE
TTHERE TO HELP YOU**

**ANSWER:**

**EXPLANATION:**

**%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%**

**close all,
clear all,
clc,**

**No_of_Random_Nummbers=1000;
Lambda = 2;
X = exprnd(Lambda,No_of_Random_Nummbers);
figure,
subplot(1,2,1); hist(X); title('Exponential Random Numbers
Histogram');
Y = icdf('Normal',X,0,1);
subplot(1,2,2); hist(Y); title('Inverse Transform Method Random
Numbers Histogram');**

**%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%**

Using MATLAB, not R codes, I repeat, please, not in R, just
MATLAB codes, write the complete code for:
1. Assume Y is an exponential random variable with rate
parameter λ=2. (1) Generate 1000 samples from this exponential
distribution using inverse transform method (2) Compare the
histogram of your samples with the true density of Y.

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

Assume that X and Y are independent random variables, each
having an
exponential density with parameter λ. Let Z = |X - Y|. What is the
density of Z?

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

1) Consider a linear congruential random number generator with
parameters a = 35, c = 20 and m = 100.
a- Generate 5 random numbers by using this method. Use 84.
b- By using inverse transform method, generate 2 random variate
for an exponential distribution with parameter λ = 0.5. Use the
first two random numbers you generated in part a.

Use R.
Generate a random sample with n=15 random observations from an
exponential distribution with mean=1.
Calculate the sample median, which is an estimator of the
population median.
Use bootstrap (nonparametric, with B=1000) methods to estimate
the variance of the estimator for the population median.
use the Monte Carlo method, e.g. generate 1000 samples of size
15 to estimate the true variance of the median estimator. Compare
and comment on your results.

how do you generate data for a y random variable in matlab?
If x has uniform distribution (0,1)
y has uniform distribution (a,b)
Then write a matlab code using rand() command to generate data
from distribution of y
y = ? + ?x
a,b, rand()

1. Suppose that X has an Exponential distribution with rate
parameter λ = 1/4. Also suppose that given X = x, Y has a
Uniform(x, x + 1) distribution.
(a) Sketch a plot representing the joint pdf of (X, Y ). Your
plot does not have to be exact, but it should clearly display the
main features. Be sure to label your axes.
(b) Find E(Y ).
(c) Find Var(Y ).
(d) What is the marginal pdf of Y ?

Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

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