Question

For each of the following functions fi(x), (i) verify that they are legitimate probability density functions (pdfs), and (ii) find the corresponding cumulative distribution functions (cdfs) Fi(t), for all t ? R.

f1(x) = |x|, ? 1 ? x ? 1

f2(x) = 4xe ?2x , x > 0

f3(x) = 3e?3x , x > 0

f4(x) = 1 2? ? 4 ? x 2, ? 2 ? x ? 2.

Answer #1

The pdf is valid since and

The CDF is for ,

The CDF is for ,

Thus the CDF is

The pdf is valid since and

The CDF is

The pdf is valid since and

Function not visible.

Write a Matlab script that plots the following functions over 0
≤ x ≤ 5π:
f1(x) = sin2 x − cos x,
f2(x) = −0.1 x 3 + 2 x 2 + 10,
f3(x) = e −x/π ,
f4(x) = sin(x) ln(x + 1).
The plots should be in four separate frames, but all four frames
should be in one figure window. To do this you can use the subplot
command to create 2 × 2 subfigures.

Consider the following functions.
f1(x) = x, f2(x) = x-1, f3(x) = x+4
g(x) = c1f1(x) + c2f2(x) + c3f3(x)
Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞).
If a nontrivial solution exists, state it. (If only the trivial
solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =?
Determine whether f1, f2, f3 are linearly independent on the
interval (−∞, ∞).
linearly dependent or linearly independent?

Let the probability density function of the random variable X be
f(x) = { e ^2x if x ≤ 0 ;1 /x ^2 if x ≥ 2 ; 0 otherwise}
Find the cumulative distribution function (cdf) of X.

Let the probability density of X be given by f(x) = c(4x - 2x^2
), 0 < x < 2; 0, otherwise. a) What is the value of c? b)
What is the cumulative distribution function of X?
c) Find P(X<1|(1/2)<X<(3/2)).

1. Find k so that f(x) is a probability density function. k=
___________
f(x)= { 7k/x^5 0 1 < x < infinity elsewhere
2. The probability density function of X is f(x).
F(1.5)=___________
f(x) = {(1/2)x^3 - (3/8)x^2 0 0 < x < 2
elsewhere
3. F(x) is the distribution function of X. Find the probability
density function of X. Give your answer as a piecewise
function.
F(x) = {3x^2 - 2x^3 0 0<x<1 elsewhere

Let X be uniformly distributed over (0, 1). Find the density
functions of the following variables by first finding their CDFs.
a. Y = 6X − 1 b. Z = X2

The following function is a legitimate density function:
f(x)=2*(3x + 4)/33 for -1 < x < 2 and 0 else. Please include
at least 3 decimal places for all parts in this question.
a) Find P(1 ≤ x < 4).
b) What is the expected value?
c) Determine the cdf.
d) Find the 50 th percentile.

Matlab
Create plot of the following density functions using x values
between -10 and 10 with an increment of 0.02
-Normal cumulative distribution function with mu=1, sigma=1,
mu=0, sigma=2, mu=0,sigma=1/2

Let X be a random variable with probability density function fX
(x) = I (0, 1) (x). Determine the probability density function of Y
= 3X + 1 and the density function of probability of Z = - log
(X).

If the probability density function of a random variable X is
ce−5∣x∣ , then (a) Compute the value of c. (b) What is the
probability that 2 < X ≤ 3? (c) What is the probability that X
> 0? (d) What is the probability that ∣X∣ < 1? (e) What is
the cumulative distribution function of X? (f) Compute the density
function of X3 . (g) Compute the density function of X2 .

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