Question

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

Answer #1

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 ≤...

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 .

Find the standard deviation of the distribution that has the
following probability density function:
f(x)={ 2x, 0<x<1 0, O.W.

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=

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 ).

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.

Let X be a continuous random variable with the probability
density function f(x) = C x, 6 ≤ x ≤ 25, zero otherwise.
a. Find the value of C that would make f(x) a valid probability
density function. Enter a fraction (e.g. 2/5): C =
b. Find the probability P(X > 16). Give your answer to 4
decimal places.
c. Find the mean of the probability distribution of X. Give your
answer to 4 decimal places.
d. Find the median...

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)

find μ and σ2 for the probability density.
For distribution function F(X):
F(x)=x^2/2 when 0<x<1
F(x)=2x-x^2/2-1 when 1<=x<2
F(x)=1 when x>=2
1.P(X>1.8) = 0.02
2.P(0.4<=X<=1.6) = 0.84

Find the values of a function f(x) if
f(x) = 2x + 5, for
f(2), f(4),
f(k), and f(x2 +
2)
The difference quotient of a function is given by:
limh0f(x)
fx+h-f(x)h . Find the
difference quotient of, f(x) = 2x + 3 and f(x) = x2 +
2
a. Prove that h(x) and k(x) are inverses of each other when
h(x) =x+213 and k(x) = 3x + 21

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