Question

The cumulative distribution function for a random variable X is F(x)= 0 if x less than or equal 0, or F(x)=sinx if 0 is less than x is less than or equal to pi/2 , or F(x)= 1, if x is greater than pi/2 . (a) find P(0.1 less than X less than 0.2. (b) find E(x)

Answer #1

Suppose that the random variable X has the following cumulative
probability distribution
X: 0 1. 2. 3. 4
F(X): 0.1 0.29. 0.49. 0.8. 1.0
Part 1: Find P open parentheses 1 less or equal than
x less or equal than 2 close parentheses
Part 2: Determine the density function f(x).
Part 3: Find E(X).
Part 4: Find Var(X).
Part 5: Suppose Y = 2X - 3, for all of X, determine
E(Y) and Var(Y)

X is a continuous random variable with the cumulative
distribution function
F(x) = 0
when x <
0
= x2
when 0 ≤ x ≤
1
= 1
when x >
1
Compute P(1/4 < X ≤ 1/2)
What is f(x), the probability density function of X?
Compute E[X]

The density function of random variable X is given by f(x) = 1/4
, if 0
Find P(x>2)
Find the expected value of X, E(X).
Find variance of X, Var(X).
Let F(X) be cumulative distribution function of X. Find
F(3/2)

A random variable X has the cumulative distribution function
(cdf) given by F(x) = (1 + e−x ) −1 , −∞ <
x < ∞.
(i) Find the probability density function (pdf) of X.
(ii) Roughly, take 10 points in the range of x (5 points below 0
and 5 points more than 0) and plot the pdf on these 10 points. Does
it look like the pdf is symmetric around 0?
(iii) Also, find the expected value of X.

6. A continuous random variable X has probability density
function
f(x) =
0 if x< 0
x/4 if 0 < or = x< 2
1/2 if 2 < or = x< 3
0 if x> or = 3
(a) Find P(X<1)
(b) Find P(X<2.5)
(c) Find the cumulative distribution function F(x) = P(X< or
= x). Be sure to define the function for all real numbers x. (Hint:
The cdf will involve four pieces, depending on an interval/range
for x....

A continuous random variable X has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise
(a) Find the value c such that f(x) is indeed
a density function.
(b) Write out the cumulative distribution function of
X.
(c) P(1 < X < 3) =?
(d) Write out the mean and variance of X.
(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of Y.

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.

Suppose a random variable X has cumulative distribution function
(cdf) F and probability
density function (pdf) f. Consider the random variable Y =
X?b
a for a > 0 and real b.
(a) Let G(x) = P(Y x) denote the cdf of Y . What is the
relationship between the functions
G and F? Explain your answer clearly.
(b) Let g(x) denote the pdf of Y . How are the two functions f
and g related?
Note: Here, Y is...

The random variable X has probability density function:
f(x) =
ke^(−x) 0 ≤ x ≤ ln 2
0 otherwise
Part a: Determine the value of k.
Part b: Find F(x), the cumulative distribution function of X.
Part c: Find E[X].
Part d: Find the variance and standard deviation of X.
All work must be shown for this question. R-Studio should not be
used.

Let X be a random variable of the mixed type having the
distribution function
F ( x ) = 0 w h e r e x < 0
F ( x ) = x 2 4 w h e r e 0 ≤ x < 1
F ( x ) = x + 1 4 w h e r e 1 ≤ x < 2
Question 1: Find the mean of X
Question 2: Find the variance of X
Question...

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