Question

1. Decide if f(x) = 1/2x^{2}dx on the interval [1, 4] is
a probability density function

2. Decide if f(x) = 1/81x^{3}dx on the interval [0, 3]
is a probability density function.

3. Find a value for k such that f(x) = kx on the interval [2, 3] is a probability density function.

4. Let f(x) = 1 /2 e ^{-x/2} on the interval [0, ∞).

a. Show that f(x) is a probability density function

b. . Find P(0 ≤ X ≤ 1)

c. Find P(X ≥ 1)

5. Find the cumulative distribution function for the probability density function

f(x) = 3 /14x^{1/2} on the interval [1, 4]

Answer #1

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)

1. f is a probability density function for the random
variable X defined on the given interval. Find the
indicated probabilities.
f(x) = 1/36(9 − x2); [−3, 3]
(a) P(−1 ≤ X ≤ 1)(9 −
x2); [−3, 3]
(b) P(X ≤ 0)
(c) P(X > −1)
(d) P(X = 0)
2. Find the value of the constant k such that the
function is a probability density function on the indicated
interval.
f(x) = kx2; [0,
3]
k=

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.

f(x)=Cx
1. what value should C be for this to be a valid probability
density function on the interval [0,4]?
2. what is the Cumulative distribution function f(x) which gives
P(X ≤ x) and use it to determine P(X ≤ 2).
3. what is the expected value of X?
4. figure out the value of E[6X+1] and Var(6X+1)

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)).

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.

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

STAT 190 Let X and Y have the joint probability density function
(PDF), f X,Y (x, y) = kx, 0 < x < 1, 0 < y < 1 -
x^2,
= 0, elsewhere,
where k is a constant.
1) What is the value of k.
2)What is the marginal PDF of X.
3) What is the E(X^2 Y).

2. Let the probability density function (pdf) of random variable
X be given by:
f(x) = C (2x -
x²),
for
0< x < 2,
f(x) = 0,
otherwise
Find the value of
C.
(5points)
Find cumulative probability function
F(x)
(5points)
Find P (0 < X < 1), P (1< X < 2), P (2 < X
<3)
(3points)
Find the mean, : , and variance,
F².
(6points)

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