Question

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)

Answer #1

Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y1, Y2, Y3, Y4, and
Y5are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
Y5are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X1, X2, X3, and X4 and
bar{Y}...

Let X1, X2, X3, and X4 be mutually independent random variables
from the same distribution. Let
S = X1 + X2 + X3 + X4. Suppose we know that S is a Chi-Square
random variable with 2 degrees of freedom. What
is the distribution of each of the Xi?

Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and variance Var(Xi) = σ2. Let V= X2−X3 and W
= X1− 2X2 + X3.
(a) Find E(V) and E(W).
(b) Find Var(V) and Var(W).
(c) Find Cov(V,W).
(d) Find the correlation coefficient ρ(V,W). Are V and W
independent?

Let X be a discrete random variable that takes on the values −1,
0, and 1. If E (X) = 1/2 and Var(X) = 7/16, what is the probability
mass function of X?

Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).
c) Compute E(Y )

Suppose we let X be a random variable that takes value 1, 2, 3
with equal probability.
Then How many samples of X are needed on average to see all
values at least once?
Also, what if X is a random variable that takes values 1, 2 or
1,2,3,4?
Thank you!

Let K be a random variable that takes, with equal probability
1/(2n+1), the integer values in the interval [-n,n].
Find the PMF of the random variable Y = In X. Where X = a^[k]. and
a is a positive number, let n = 7 and a = 2. Then what is E[Y
]?

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

A uniform random variable on (0,1), X, has density function f(x)
= 1, 0 < x < 1. Let Y = X1 + X2 where X1 and X2 are
independent and identically distributed uniform random variables on
(0,1).
1) By considering the cumulant generating function of Y ,
determine the first three cumulants of Y .

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