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.

The probability density function for a continuous random
variable X is given by
f(x) =
0.6 0<X<1
=
0.10(x) 1 ≤X≤ 3
=
0 otherwise
Find the 85th percentile value of X.

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)

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....

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. 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 ≤...

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).]

If the probability density function of a random variable X is
ce−5∣x∣ , then (a) Compute the value of c. (b) What is the
probability that 2 < X ≤ 3? (c) What is the probability that X
> 0? (d) What is the probability that ∣X∣ < 1? (e) What is
the cumulative distribution function of X? (f) Compute the density
function of X3 . (g) Compute the density function of X2 .

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)

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