Question

Question 7) Suppose X is a Normal random variable with with
expected value 31 and standard deviation 3.11. We take a random
sample of size n from the distribution of X. Let X be the sample
mean. Use R to determine the following:

a) Find the probability P(X>32.1):

b) Find the probability P(X >32.1) when n = 4:

c) Find the probability P(X >32.1) when n = 25:

d) What is the probability P(31.8 <X <32.5) when n =
25?

e) What is the standard deviation of X when n = 19?

f) What is the probability that X_{1} + X_{2} + ...
+X_{20} > 630?

g) Copy your R script for the above into the text box here.

Answer #1

Suppose X is a Normal random variable with with expected value
31 and standard deviation 3.11. We take a random sample of size n
from the distribution of X. Let X be the sample mean. Use R
to determine the following:
a) What is the standard deviation of X when n =
19?
b) What is the probability that X1 + X2 +
... +X20 > 630?
PLEASE ANSWER IN R SCRIPT

Suppose X is a random variable with with expected value -0.01
and standard deviation σ = 0.04.
Let
X1,
X2, ...
,X81
be a random sample of 81 observations from the distribution of
X.
Let X be the sample mean. Use R to determine the
following:
Copy your R script
b) What is the approximate probability that
X1 +
X2 + ...
+X81 >−0.02?

Problem #3. X is a random variable with an exponential
distribution with rate λ = 7 Thus the pdf of X is f(x) = λ
e−λx for 0 ≤ x where λ = 7.
a) Using the f(x) above and the R integrate function calculate the
expected value of X.
b) Using the f(x) above and the R integrate function calculate the
expected value of X2
c) Using the dexp function and the R integrate command calculate
the expected value...

Question 6) Suppose X is a random variable taking on possible
values 0,2,4 with respective probabilities .5, .3, and .2. Y is a
random variable independent from X taking on possible values 1,3,5
with respective probabilities .2, .2, and .6. Use R to determine
the following.
f) Find the expected value of X*Y. (i.e. X times Y)
g) Find the expected value of 3X - 5Y.
h) Find the variance of 3X - 5Y
i) Find the expected value of...

Question 2) The density of random variable X is f(x) =
15(x2−36)(64−x2) / 3904 for 6 ≤ x ≤ 8 and 0
otherwise. Do computations using the R integrate function.
a) Find the probability that X > 7:
b) Find the probability that 6.5 < X < 7.5:
e) Find the probability that x is within one standard deviation
of its expected value:
f) In the following paste your R script for this problem:

Suppose that Xi are IID normal random variables with
mean 5 and variance 10, for i = 1, 2, ... , n.
(a) Calculate P(X1 < 6.2), i.e., the probability that
the first value collected is less than 6.2.
(b) Suppose we collect a sample of size 2, X1 and
X2. What is the probability that their sample mean is
greater than 6.3?
(c) Again, suppose we collect two samples (n=2), X1 and
X2. What is the probability that their...

Suppose a random variable X takes on the value of -1 or 1, each
with the probability of 1/2. Let y=X1+X2+X3+X4, where X1,....X4 are
independent. Find E(Y) and Find Var(Y)

Let the random variable X follow a distribution with a mean of
μ and a standard deviation of σ. Let X1 be the mean of a sample of
n1 (n1=1) observations randomly chosen from
this population, and X2 be the mean of a sample of n2(
n2 =49) observations randomly chosen from the same
population. Which of the following statement is False? Evaluate the
following statement.
P(μ
- 0.2σ <X 1 < μ + 0.2σ) <
P(μ - 0.2σ <X...

Consider independent random variables X and Y , such that X has
mean 2 and standard deviation 4, and Y has mean 1 and standard
deviation 9. Find the mean and standard deviation of the following
random variables. a) 3X b) Y + 6 c) X + Y d) X − Y e) X1 + X2,
where X1, X2 are independent copies of X.

Let X1, X2 be a random sample of size 2 from the standard normal
distribution N (0, 1). find the distribution of {min(X1, X2)}^2

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