9. Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Find the expected value of the length of the longer piece.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
question 9
![8. Suppose that X and Y have a continuous joint distribu-
tion for which the joint p.d.f. is as follows:
f(x, y) = { 12y- for 0 < y < x < 1,
otherwise.
Find the value of E(XY).
9. Suppose that a point is chosen at random on a stick of
unit length and that the stick is broken into two pieces at
that point. Find the expected value of the length of the
longer piece.
10. Suppose that a particle is released at the origin of
the xy-plane and travels into the half-plane where x > 0.
Suppose that the particle travels in a straight line and that
the angle between the positive half of the x-axis and this
line is a, which can be either positive or negative. Suppose,
finally, that the angle a has the uniform distribution on the
interval [-x/2, x/2]. Let Y be the ordinate of the point at
which the particle hits the vertical line x = 1. Show that
the distribution of Y is a Cauchy distribution.
11. Suppose that the random variables X1, .
a random sample of size n from the uniform distribution
on the interval [0, 1]. Let Y1
Y, = max{X1, ..., X,}. Find E(Y1) and E (Y„).
form
min{X1, .
X,}, and let
12. Suppose that the random variables X1,..., X, form
a random sample of size n from a continuous distribution
for which the c.d.f. is F, and let the random variables Y,
and Y, be defined as in Exercise 11. Find E[F(Yj)] and
E[F(Y„)].
13. A stock currently sells for $110 per share. Let the price
of the stock at the end of a one-year period be X, which will
take one of the values $100 or $300. Suppose that you have
the option to buy shares of this stock at $150 per share
at the end of that one-year period. Suppose that money
4.2 Properties of Expectations
217
231
c. Consider the same transactions as in part (a), but
this time suppose that the option price is $x where
x > 20.19. Prove that our investor gains 4.16x – 84
dollars of net worth no matter what happens to the
stock price.
911
The situations in parts (b) and (c) are called arbi-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ceeea51-465f-4606-b567-3c32217ff55b%2F00f53c7f-5620-4ba1-8a30-3852d5db2688%2Fv2eswco_processed.jpeg&w=3840&q=75)
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