Question

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)

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 ≤...

A random variable X representing the time (in hours) a furnace
is running is drawn from Ω = [0, 4] according to the density
function f(x) = Cx.
(a) What value must the constant C be in order for f to be a
valid probability density function?
(b) Determine the cdf F(x) which gives P(X ≤ x) and use it to
determine P(X ≤ 2).
(c) Determine the expected value of X
. (d) Determine E[6X + 1] and Var(6X...

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 X be a random variable with probability density function
fX(x) = {c(1−x^2)if −1< x <1, 0 otherwise}.
a) What is the value of c?
b) What is the cumulative distribution function of X?
c) Compute E(X) and Var(X).

Suppose you choose a real number X from the interval
[3,16] with the density function
f(x)=Cx,
where C is a constant.
a) Find C. Remember that if you integrate a density
function over the entire sample space interval, you should get
1.
b) Find P(E), where
E=[a,b] is a subinterval of [3,16]
(as a function of a and b ).
c) Find P(X>4)
d) Find P(X<14)
e) Find P(X^2−18X+56≥0)
Note: You can earn partial credit on this
problem.

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...

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.

Suppose a random variable has the following probability density
function: f(x)=3cx^2 (1-x) 0≤x≤1
a) What must c be equal to for this to be a valid density
function?
b) Determine the mean of x, μ_x
c) Determine the median of x, μ ̃_x
d) Determine: P(0≤x≤0.5) ?

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.

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=

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