Question

A continuous random variable *X* has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise

(a) Find the value *c* such that *f(x)* is indeed
a density function.

(b) Write out the cumulative distribution function of
*X*.

(c) *P(1 < X < 3) =?*

(d) Write out the mean and variance of *X*.

(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of *Y*.

Answer #1

suppose x is a continuous random variable with probability
density function f(x)= (x^2)/9 if 0<x<3 0 otherwise
find the mean and variance of x

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

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.

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....

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)

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

1. Suppose a random variable X has a probability density
function
f(x)= {cx^2 -1<x<1,
{0 otherwise
where c > 0.
(a) Determine c.
(b) Find the cdf F ().
(c) Compute P (-0.5 < X < 0.75).
(d) Compute P (|X| > 0.25).
(e) Compute P (X > 0.75 | X > 0).
(f) Compute P (|X| > 0.75| |X| > 0.5).

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...

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.

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