Question

find the value of k for which the given funcion is a probability
density funcion over the interval

f(x)= 4x on [1, k]

Answer #1

**X is a continuous random variable, whose probability
density function is given by**

**
When .**

**We have to find the value of k.**

**Now, we know that a probability density function, when
integrated over the whole range of the random variable, results in
1.**

**So, by this property,**

**So, approximately, k is 1.2247.**

**Thus, the value of k, for which the given function is a
probability density function, is square root of 1.5, ie. 1.2247
approximately.**

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=

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

If f(x,y) = k is a joint probability density function over the
region 0<x<4, 0<y, and x-1<y<x+1, what is the value
of f(x)?

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 ≤...

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)).

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 average value of the function over the given interval
and all values of x in the interval for which the function
equals its average value. (Round your answer to three decimal
places.)
f(x) = 4x3 − 3x2, [−1,
3]
(x, y) =

Given the following probability distribution for the random
variable X, find the value of k, then find the mean (expected
value) of X. Do not round your answers. X = 0,1,2,3..... P(X=x) :
k,.35,.2,.15

The probability density function of X is given by
f(x)={a+bx0if 0<x<1otherwise
If E(X)=1.5, find a+b.
Hint: For a probability density function f(x), we have
∫∞−∞f(x)dx=1

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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