If X follows the following probability distribution:
.20 2 <X <3
f (x) = .60 3 <X <4
.20 4 <X <5
0 for other X’s
a. Calculate the cumulative probability function of X and make a
reasonable graphical representation. (15 pts)
b. Calculate the expected value of X and the Variance of X. (15
pts)
c. Calculate the probability that X is between 2.40 and 3.80. (10
pts)
d. Calculate the percentile of 70 percent. (10 pts)
e. If...
Let be the following probability density function f (x) = (1/3)[
e ^ {- x / 3}] for 0 <x <1 and f (x) = 0 in any other
case
a) Determine the cumulative probability distribution F (X)
b) Determine the probability for P (0 <X <0.5)
a. For the following probability density
function:
f(X)=
3/4 (2X-X^2 ) 0 ≤ X ≤ 2
=
0 otherwise
find
its expectation and variance.
b. The two regression lines are 2X - 3Y + 6 = 0 and 4Y – 5X- 8
=0 , compute mean of X and mean of Y. Find correlation coefficient
r , estimate y for x =3 and x for y = 3.
Suppose X has probability distribution
x: 0 1 2 3 4
P(X = x) 0.2 0.1 0.2 0.2 0.3
Find the following probabilities:
a. P(X < 2)
b. P(X ≤ 2 and X < 4)
c. P(X ≤ 2 and X ≥ 1)
d. P(X = 1 or X ≤ 3)
e. P(X = 2 given X ≤ 2)
x
0
1
2
3
4
P(X)
.45
.3
.2
.04
.01
(c) Find the probability that a person has 1 sibling given that
they have less than 3 siblings. Hint: Use the conditional formula:
P(A|B)=P(A and B)/P(B). In this case A: event of having 1 sibling
and B: event of having less than 3 siblings.
(d) Find the probability that a person has at least 1 sibling OR
less than 2 siblings. Hint: Use the General Addition Rule: P(A...
Let Poly3(x) = polynomials in x of degree at most 2. They form a
3- dimensional space. Express the operator T(p) = p'
as a matrix (i) in basis {1, x, x 2 }, (ii) in basis {1, x, 1+x
2 } .
Let f(x) = b(x+1), x = 0, 1, 2, 3 be the probability mass
function (pmf) of a random variable X, where b is constant.
A. Find the value of b
B. Find the mean μ
C. Find the variance σ^2
Question 3
Suppose that X and Y have the following joint probability
distribution:
f(x,y)
x
0
1
2
y
0
0.12
0.08
0.06
1
0.04
0.19
0.12
2
0.04
0.05
0.3
Find the followings:
E(Y)=
Var(X)=
Cov(X,Y)=
Correlation(X,Y)=
The (mixed) random variable X has probability density function
(pdf) fX (x) given by:
fx(x)=0.5δ(x−3)+ { c.(4-x2), 0≤x≤2
0, otherwise
where c is a constant.
(a) Sketch fX (x) and find the constant c.
(b) Find P (X > 1).
(c) Suppose that somebody tells you {X > 1} occurred. Find
the conditional pdf fX|{X>1}(x), the pdf of X given
that {X > 1}.
(d) Find FX(x), the cumulative distribution function of X.
(e) Let Y = X2 . Find...