Question

*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 − x^{2}); [−3, 3]

(a) *P*(−1 ≤ *X* ≤ 1)(9 −
x^{2}); [−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*) = *kx*^{2}; [0,
3]

k=

Answer #1

Probability density function of the continuous random variable X
is given by f(x) = ( ce −1 8 x for x ≥ 0 0 elsewhere
(a) Determine the value of the constant c.
(b) Find P(X ≤ 36).
(c) Determine k such that P(X > k) = e −2 .

1 (a) Let f(x) be the probability density function of a
continuous random variable X defined by
f(x) = b(1 - x2), -1 < x < 1,
for some constant b. Determine the value of b.
1 (b) Find the distribution function F(x) of X . Enter the value
of F(0.5) as the answer to this question.

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)

1. Decide if f(x) = 1/2x2dx on the interval [1, 4] is
a probability density function
2. Decide if f(x) = 1/81x3dx 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 ≤...

7. For the random variable x with probability density function:
f(x) = {1/2 if 0 < x< 1, x − 1 if 1 ≤ x < 2}
a. (4 points) Find the CDF function. b. (3 points) Find p(x <
1.5). c. (3 points) Find P(X<0.5 or X>1.5)

1. Let X be a discrete random variable with the probability mass
function P(x) = kx2 for x = 2, 3, 4, 6.
(a) Find the appropriate value of k.
(b) Find P(3), F(3), P(4.2), and F(4.2).
(c) Sketch the graphs of the pmf P(x) and of the cdf F(x).
(d) Find the mean µ and the variance σ 2 of X. [Note: For a
random variable, by definition its mean is the same as its
expectation, µ = E(X).]

Let X be a random variable with probability density function
f(x) = {3/10x(3-x) if 0<=x<=2
.........{0 otherwise
a) Find the standard deviation of X to four decimal
places.
b) Find the mean of X to four decimal places.
c) Let y=x2 find the probability density function
fy of Y.

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.

Consider a continuous random variable X with the probability
density function f X ( x ) = |x|/C , – 2 ≤ x ≤ 1, zero elsewhere.
a) Find the value of C that makes f X ( x ) a valid probability
density function. b) Find the cumulative distribution function of
X, F X ( x ).

Q6/
Let X be a discrete random variable defined by the
following probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Give P(4≤ X < 8)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q7/
Let X be a discrete random variable defined by the following
probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Let F(x) be the CDF of X. Give F(7.5)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q8/
Let X be a discrete random variable defined by the following
probability function :
x
2
6...

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