Question

Solve the following:

(a)A sample of 3 items is chosen at random from a box
containing 20 items of which 4 are

defective. Let X be the number of defective items in the
sample. Find E(X), Var (X), and Var (20 − X).

(b) Given n tosses of a fair coin, let X be the number of
heads tossed. Find the pmf of X

(c) Show that Var (X) ≥ 0 for any discrete r.v. X. Also,
explain why X is constant if and only if

Var (X) = 0. Hint: write Var (X) = E (X − µ)2 = Px (x − µ)^2pX
(x), where µ = E (X)

Answer #1

Complete the following:
(a) Var (X) ≥ 0 for any discrete r.v. X. Also, explain why X
is constant if and only if
Var (X) = 0. Hint: write Var (X) = E (X − µ)2 = Px (x − µ)2 pX
(x), where µ = E (X).
(b) Let X represent the dierence between the number of heads
and the number of tails obtained
when a fair coin is tossed n times. Find E (X) and Var
(X)

A coin is tossed 5 times. Let the random variable ? be the
difference between the number of heads and the number of tails in
the 5 tosses of a coin. Assume ?[heads] = ?.
Find the range of ?, i.e., ??.
Let ? be the number of heads in the 5 tosses, what is the
relationship between ? and ?, i.e., express ? as a function of
??
Find the pmf of ?.
Find ?[?].
Find VAR[?].

A box contains 7 items, 4 of which are defective. a random
sample of 3 items are taken from the box. Let X be the number of
defective items in the sample. 1.Find the probability mass function
of X. 2.Find the mean and the variance of X.

A fair coin has been tossed four times. Let X be the number of
heads minus the number of tails (out of four tosses). Find the
probability mass function of X. Sketch the graph of the probability
mass function and the distribution function, Find E[X] and
Var(X).

NOTE:KINDLY SOLVE PARTS D AND E.
A fair coin is tossed four times, and the random variable X is
the number of heads in the first three tosses and the random
variable Y is the number of heads in the last three tosses.
a) What is the joint probability mass function of X and Y ?
b) What are the marginal probability mass functions of X and Y
?
c) Are the random variables X and Y independent?
d) What...

A fair coin is tossed three times. Let X be the number of heads
among the first two tosses and Y be the number of heads among the
last two tosses. What is the joint probability mass function of X
and Y? What are the marginal probability mass function of X and Y
i.e. p_X (x)and p_Y (y)? Find E(X) and E(Y). What is Cov(X,Y) What
is Corr (X,Y) Are X and Y independent? Explain. Find the
conditional probability mass...

Let X be a number chosen at random from the set
{1, 2, ... ,20} and let Y be a number chosen at random
from the set {1, 2, ... , X }. Let
pX |Y (x|y)
denote the condition distribution of X, given that
Y = y. Find
pX |Y (19|18)

I toss 3 fair coins, and then re-toss all the ones that come up
tails. Let X denote the number of coins that come up heads on the
first toss, and let Y denote the number of re-tossed coins that
come up heads on the second toss. (Hence 0 ≤ X ≤ 3 and 0 ≤ Y ≤ 3 −
X.)
(a) Determine the joint pmf of X and Y , and use it to calculate
E(X + Y )....

From a box of fruit containing 55 oranges and 1 apple a random
sample of 2 pieces of fruit has been selected without replacement.
Let X be the number of oranges and Y be the number of apples in the
sample. What will the covariance of X and Y be?

From a box of fruit containing 82 oranges and 1 apple a random
sample of 2 pieces of fruit has been selected without replacement.
Let X be the number of oranges and Y be the number of apples in the
sample. What will the Variance of Y be?

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