Question

A number x is selected at random in the interval [-1, 2]. Let the events A = {x<0} and B = {|x-0.5|<0.5}, and C = {x >0.75} (a) Find the probabilities of A, B, AB and AC. (b) Find the probabilities of AuB, AuC and AuBuC by using the appropriate axioms or corollaries.

Answer #1

The pdf of X is

(a)

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Now,

So,

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(b)

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A number x is selected at random in the interval [-3,3]. Let the
events A={x < 0}, B={|x - 0.5| < 0.5|}, and C={x > 0.75}.
Find P[A|B], P[B|C],P[A|CC],P[B|CC].

Let X be a randomly selected real number from the interval [0,
1]. Let Y be a randomly selected real number from the interval [X,
1].
a) Find the joint density function for X and Y.
b) Find the marginal density for Y.
c) Does E(Y) exist? Explain without calculation. Then find
E(Y).

Let X be a random number between 0 and 1 produced by a random
number generator. The random number generator will spread its
output uniformly (evenly) across the entire interval from 0 to 1.
All numbers have an equal probability of being selected. Find the
value of aa that makes the following probability statements
true.
(a) P(x≤a)=0.8
a=
(b) P(x < a) = 0.25
a=
(c) P(x≥a)=0.17
a=
(d) P(x>a)=0.73
a=
(e) P(0.15≤x≤a)=
a=

2. X123456
Suppose that the random variable, X, is a number on the biased
die and the p.d.f. of X is as shown below;
P(X=x) 1/6 1/6 1/5 k
a) Find;
(i) the value of k.
(ii) E(X)
(iii) E(X2)
(iv) V ar(X)
(v) P(1£X<5)
1/5 1/6
b) If events A and B are such that they are independent, and
P(A) = 0.3 with P(B) = 0.5;
Find P(A n B) and P(AUB)
Are A and B mutually...

Let A1, ... ,20 be independent events each with probability 1/2.
Let X be the number of events among the first 10 which occur and
let Y be the number of events among the last 10 which occur. Find
the conditional probability that X = 5, given that X + Y = 12.

1. Suppose a random variable X has a probability density
function
f(x)= {cx^2 -1<x<1,
{0 otherwise
where c > 0.
(a) Determine c.
(b) Find the cdf F ().
(c) Compute P (-0.5 < X < 0.75).
(d) Compute P (|X| > 0.25).
(e) Compute P (X > 0.75 | X > 0).
(f) Compute P (|X| > 0.75| |X| > 0.5).

Let X represent a continuous random variable with a Uniform
distribution over the interval from 0 to 2. Find the following
probabilities (use 2 decimal places for all answers): (a) P(X ≤
1.92) = (b) P(X < 1.92) = (c) P(0.22 ≤ X ≤ 1.56) = (d) P(X <
0.22 or X > 1.56) =

One number is randomly selected from the following set: { 1, 2,
3, 4, 5, 6, 7, 8, 9, 10 }.
Let
A = event
that the selected number is even
B = event that the selected number is a multiple of 3
Find the following
probabilities.
a) P( A
and B
b) P( A
or B
)
c) P(
A B)
d) Are events A...

Let A and B be independent events of some sample space. Using
the definition of independence P(AB) = P(A)P(B), prove that the
following events are also independent:
(a) A and Bc
(b) Ac and B
(c) Ac and Bc

3. Let X be a continuous random variable with PDF
fX(x) = c / x^1/2, 0 < x < 1.
(a) Find the value of c such that fX(x) is indeed a PDF. Is this
PDF bounded?
(b) Determine and sketch the graph of the CDF of X.
(c) Compute each of the following:
(i) P(X > 0.5).
(ii) P(X = 0).
(ii) The median of X.
(ii) The mean of X.

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