Question

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.

Answer #1

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.

Part A
The variable X(random variable) has a density function of the
following
f(x) = {5e-5x if 0<= x < infinity and 0
otherwise}
Calculate E(ex)
Part B
Let X be a continuous random variable with probability density
function
f (x) = {6/x2 if 2<x<3 and 0 otherwise }
Find E (ln (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 the random variable with probability density function
f(x) = 0.5x for 0 ≤ x ≤ 2 and zero otherwise. Find the
mean and standard deviation of the random variable X.

Let X be a random variable with probability density function
f(x) = { λe^(−λx) 0 ≤ x < ∞
0 otherwise } for some λ > 0.
a. Compute the cumulative distribution function F(x), where F(x)
= Prob(X < x) viewed as a function of x.
b. The α-percentile of a random variable is the number mα such
that F(mα) = α, where α ∈ (0, 1). Compute the α-percentile of the
random variable X. The value of mα will...

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

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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, 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)

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 random variable X has a probability density function f(x) =
e^(−x) for x > 0. If a > 0 and A is the event that X > a,
find f XIA (xlx > a), i.e. the density of the conditional
distribution of X given that X > a.

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