Question

X is a discrete random variable representing number of conforming parts in a sample and has following probability mass function

?(?) = { ?(5 − ?) if ? = 1, 2, 3, 4

0 otherwise

i) Find the value of constant ? and justify your answer

. ii) ( Determine the cumulative distribution function of X, (in the form of piecewise function).

iii) Use the cumulative distribution function found in question 2 to determine the following:

a) ?(2 < ? ≤ 4)

b) ?(? ≥ 3)

Answer #1

Let X be the random variable representing the difference between
the number of headsand the number of tails obtained when a fair
coin is tossed 4 times.
a) What are the possible values of X?
b) Compute all the probability distribution of X?
c) Draw the cumulative distribution function F(x) of the random
variable X.

Suppose that the probability mass function for a discrete random
variable X is given by p(x) = c x, x = 1, 2, ... , 9. Find the
value of the cdf (cumulative distribution function) F(x) for 7 ≤ x
< 8.

Find the joint discrete random variable x and y,their joint
probability mass function is given by Px,y(x,y)={k(x+y)
x=-2,0,+2,y=-1,0,+1
0 Otherwise }
2.1 determine the value of constant k,such that this will be
proper pmf?
2.2 find the marginal pmf’s,Px(x) and Py(y)?
2.3 obtain the expected values of random variables X and Y?
2.4 calculate the variances of X and Y?

Let X be a random variable with probability density function
fX(x) given by fX(x) = c(4 − x ^2 ) for |x| ≤ 2 and zero
otherwise.
Evaluate the constant c, and compute the cumulative distribution
function.
Let X be the random variable. Compute the following
probabilities.
a. Prob(X < 1)
b. Prob(X > 1/2)
c. Prob(X < 1|X > 1/2).

If X is a discrete random variable with uniform distribution,
where f(x) > 0 when x = -1, 0, and 1(f(x) = 0, otherwise). If Y
is another discrete random variable with identical distribution as
X. In addition, X and Y are independent.
1. Please find the probability distribution of (X + Y)/2 and
plot it.
2. Please find variance of (X + Y)/2

Let X be a discrete random variable with probability mass
function (pmf) P (X = k) = C *ln(k) for k = e; e^2 ; e^3 ; e^4 ,
and C > 0 is a constant.
(a) Find C.
(b) Find E(ln X).
(c) Find Var(ln X).

Q6/
Let X be a discrete random variable defined by the
following probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Give P(4≤ X < 8)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q7/
Let X be a discrete random variable defined by the following
probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Let F(x) be the CDF of X. Give F(7.5)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q8/
Let X be a discrete random variable defined by the following
probability function :
x
2
6...

Problem 3. Let x be a discrete random variable with the
probability distribution given in the following table:
x = 50 100 150 200 250 300 350
p(x) = 0.05 0.10 0.25 0.15 0.15 0.20 0.10
(i) Find µ, σ 2 , and σ.
(ii) Construct a probability histogram for p(x).
(iii) What is the probability that x will fall in the interval
[µ − σ, µ + σ]?

Consider the following question and determine if the random
variable X is Continuous or
Discrete. Select a distribution to use
from the options to find the exact probability in question. Select
either Uniform, Binomial,
Poisson, Normal, or
Exponential. Do not solve the problem simply
explain how you arrived at your conclusion when selecting the
random variable and the selected distribution.
Question:
You have taken a sample from a batch of parts and have averaged
the diameter of the parts in...

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.

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