Question

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(e ^{x})**

Part B

Let X be a continuous random variable with probability density function

f (x) = {6/x^{2} if 2<x<3 and 0 otherwise }

**Find E (ln (X)).**

.

Answer #1

Part A

for

Part B

for 2 < x < 3

Let t = ln(x) then range of t is ln(2) to ln(3)

dt = dx/x

and x = e^t

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.

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.

Let X be a random variable with density function f(x) = 2 5 x
for x ∈ [2, 3] and f(x) = 0, otherwise. (a) (6 pts) Compute E[(X −
2)3 ] without attempting to find the density function of Y = (X −
2)3 . (b) (6 pts) Find the density function of Y = (X − 2)3

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.

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 following
probability density function:
f(x) = e^−(x−1) for x ≥ 1; 0 elsewhere
(i) Find P(0.5 < X < 2).
(ii) Find the value such that random variable X exceeds it 50%
of the time. This value is called the median of the random variable
X.

If f(x) is a probability density function of a continuous random
variable, then f(x)=?
a-0
b-undefined
c-infinity
d-1

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.

Let X1,X2,...,X50 denote a random sample of size 50 from the
distribution whose probability density function is given by f(x)
=(5e−5x, if x ≥ 0 0, otherwise If Y = X1 + X2 + ... + X50, then
approximate the P(Y ≥ 12.5).

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.

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