Let X be a continuous random variable such that E[Xm]
exists where m is some positive...
Let X be a continuous random variable such that E[Xm]
exists where m is some positive integer. Prove that if k is a
positive integer and k < m, then E[Xk] exists.
Does a continuous random variable X exist with E(X − a) = 0,
where a is the
day of your birthdate? If yes, give an example for its probability
density function.
If no, give an explanation.
(b) Does a continuous random variable Y exist with E
(Y − b)
2
= 0, where b is
the month of your birthdate? If yes, give an example for its
probability density
function. If no, give an explanation.
(a) Let X be a continuous random variable which only takes on
positive values on the interval [1, 4]. If P(X) = (√ x + √ 1 x )C 2
for all x in this interval, compute the value of C.
(b) Let X be a random variable with normal distribution. Let z
represent the z-score for X, and let a be a positive number. Prove
that P(z < |a|) = P(z < a) + P(z > −a) − 1.
Let X be a random variable with CDF F(x) =
e-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞.
1. What is the median of X?
2. Obtain the PDF of X. Use R to plot, in the range
-10<x<30, the pdf for µ = 2, β = 5.
3. Draw a random sample of size 1000 from f(x) for µ = 2, β = 5
and draw a histogram of the values in the random sample...
Let X and Y be continuous random variables with E[X] = E[Y] = 4
and var(X) = var(Y) = 10. A new random variable is defined as: W =
X+2Y+2. a. Find E[W] and var[W] if X and Y are independent. b. Find
E[W] and var[W] if E[XY] = 20. c. If we find that E[XY] = E[X]E[Y],
what do we know about the relationship between the random variables
X and Y?
1. Let X be a continuous random variable such
that when x = 10, z = 0.5. This z-score tells us that x = 10 is
less than the mean of X.
Select one:
True
False
2. If an economist wants to determine if there
is evidence that the average household income in a community is
different from $ 32,000, then a two-tailed hypothesis test should
be used.
Select one:
True
False
3. α (alpha) refers to the proportion of...
Let x be a continuous random variable that follows a
distribution skewed to the left with ?= 92 and ?=15. Assuming n/N
<= .05, find the probability that the sample mean, x bar, for a
random sample of 62 taken from this population will be (ROUND
ANSWERS TO FOUR DECIMAL PLACES):
a) less than 81.5
P(less that 81.5)=
b) greater than 89.7
P(greater than 89.7)=
Please show your work.
Let x be a continuous random variable
that is normally distributed with a mean of 65 and a standard
deviation of 15. Find the probability that
x assumes a value:
less than 48
greater than 87
between 56 and 70
Let X be a continuous random variable with a probability density
function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) =
e −x
a. Find the expression for the probability density function fY
(y).
b. Find the domain of the probability density function fY
(y).
2] Let x be a continuous random variable that has a normal
distribution with μ = 48 and σ = 8 . Assuming n N ≤ 0.05 , find the
probability that the sample mean, x ¯ , for a random sample of 16
taken from this population will be between 49.64 and 52.60 .
Round your answer to four decimal places.
Let x be a continuous random variable that has a normal
distribution with μ = 60 and σ = 12. Assuming
n ≤ 0.05N, where n = sample size and
N = population size, find the probability that the sample
mean, x¯, for a random sample of 24 taken from this population will
be between 54.91 and 61.79.
Let x be a continuous random variable that has a normal
distribution with μ = 60 and σ = 12. Assuming
n...