Question

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)

Answer #1

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)

Suppose X is a discrete random variable with probability mass
function given by
p (1) = P (X = 1) = 0.2
p (2) = P (X = 2) = 0.1
p (3) = P (X = 3) = 0.4
p (4) = P (X = 4) = 0.3
a. Find E(X^2) .
b. Find Var (X).
c. Find E (cos (piX)).
d. Find E ((-1)^X)
e. Find Var ((-1)^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)

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Question 1
Refer to the probability function given in the following table
for a
random variable X that takes on the values 1,2,3 and 4
X 1 2 3 4
P(X=x) 0.4 0.3 0.2 0.1
a) Verify that the above table meet the conditions
for being a discrete probability
distribution
b) Find P(X<2)
c) Find P(X=1 and X=2)
d) Graph P(X=x)
e) Calculate the mean of the random variable
X
f) Calculate the standard deviation of the random
variable X...

Suppose that a random variable X has the distribution (pdf) f(x)
=kx(1 -x^2) for
0 < x < 1 and zero elsewhere.
a. Find k.
b. Find P(X >0. 8)
c. Find the mean of X.
d. Find the standard deviation of X.
2. Assume that test scores for all students on a statistics test
are normally
distributed with mean 82 and standard deviation 7.
a. Find the probability that a single student scores greater than
80.
b. Find the...

Suppose we have the following probability mass function.
X
0
2
4
6
8
F(x)
0.1
0.3
0.2
0.3
0.1
a) Determine the cumulative distribution function, F(x).
b) Determine the expected value (mean), E(X) = μ.
c) Determine the variance, V(X) = σ^2

Consider the following cumulative probability
distribution.
x
0
1
2
3
4
5
P(X ≤ x)
0.10
0.29
0.48
0.68
0.84
1
a. Calculate P(X ≤ 2).
(Round your answer to 2 decimal places.)
b. Calculate P(X = 2).
(Round your answer to 2 decimal places.)
c. Calculate P(2 ≤ X ≤ 4).
(Round your answer to 2 decimal places.)

Consider the following joint distribution between random
variables X and Y:
Y=0
Y=1
Y=2
X=0
P(X=0, Y=0) = 5/20
P(X=0, Y=1) =3/20
P(X=0, Y=2) = 1/20
X=1
P(X=1, Y=0) = 3/20
P(X=1, Y=1) = 4/20
P(X=1, Y=2) = 4/20
Further, E[X] = 0.55, E[Y] = 0.85, Var[X] = 0.2475 and Var[Y] =
0.6275.
a. (6 points) Find the covariance between X and Y.
b. (6 points) Find E[X | Y = 0].
c. (6 points) Are X and Y independent?...

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.

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