Problem # 4. Yet another variation on a vaguely familiar theme...: A startup has developed a questionable AI-based app for automatic trading on the stock market. The following probabilistic model is used to analyze and predict the daily gain/loss performance of the app. Let random variable R represent the anticipated range of fluctuation (gain or loss) in the value of the managed stock portfolio during the day, measured in millions of dollars, whose PDF is fR(r) = re¯u(r). Let random variable X be the actual gain/loss in value by the end of the day. It was observed that when R takes the value r, X is distributed uniformly in the range [—r,r], i.e., it ranges between a loss and a gain of r million dollars. a) Find the marginal density of the actual gain/loss, fx(x). b) Are R and X uncorrelated, orthogonal and/or independent? c) Having observed the actual gain/loss of the portfolio at the end of the day, it is desirable to update the model for the range R. Find the conditional PDF fRx(r|x). d) An optional safety measure is offered to avoid excessive losses. It stops further trading for the day whenever the actual loss reaches a certain limit, namely, a loss of c million dollars. Thus, with the safety measure in place, the actual daily gain/loss is represented by random variable Y: -c X <-c, Y = X X >-c Find the PDF of Y. (4)
Problem # 4. Yet another variation on a vaguely familiar theme...: A startup has developed a questionable AI-based app for automatic trading on the stock market. The following probabilistic model is used to analyze and predict the daily gain/loss performance of the app. Let random variable R represent the anticipated range of fluctuation (gain or loss) in the value of the managed stock portfolio during the day, measured in millions of dollars, whose PDF is fR(r) = re¯u(r). Let random variable X be the actual gain/loss in value by the end of the day. It was observed that when R takes the value r, X is distributed uniformly in the range [—r,r], i.e., it ranges between a loss and a gain of r million dollars. a) Find the marginal density of the actual gain/loss, fx(x). b) Are R and X uncorrelated, orthogonal and/or independent? c) Having observed the actual gain/loss of the portfolio at the end of the day, it is desirable to update the model for the range R. Find the conditional PDF fRx(r|x). d) An optional safety measure is offered to avoid excessive losses. It stops further trading for the day whenever the actual loss reaches a certain limit, namely, a loss of c million dollars. Thus, with the safety measure in place, the actual daily gain/loss is represented by random variable Y: -c X <-c, Y = X X >-c Find the PDF of Y. (4)
A First Course in Probability (10th Edition)
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ISBN:9780134753119
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Chapter1: Combinatorial Analysis
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![Problem # 4. Yet another variation on a vaguely familiar theme...:
A startup has developed a questionable AI-based app for automatic trading on the stock
market. The following probabilistic model is used to analyze and predict the daily gain/loss
performance of the app. Let random variable R represent the anticipated range of fluctuation
(gain or loss) in the value of the managed stock portfolio during the day, measured in millions
of dollars, whose PDF is fR(r) = re¯u(r). Let random variable X be the actual gain/loss in
value by the end of the day. It was observed that when R takes the value r, X is distributed
uniformly in the range [—r,r], i.e., it ranges between a loss and a gain of r million dollars.
a) Find the marginal density of the actual gain/loss, fx(x).
b) Are R and X uncorrelated, orthogonal and/or independent?
c) Having observed the actual gain/loss of the portfolio at the end of the day, it is desirable to
update the model for the range R. Find the conditional PDF fRx(r|x).
d) An optional safety measure is offered to avoid excessive losses. It stops further trading for
the day whenever the actual loss reaches a certain limit, namely, a loss of c million
dollars. Thus, with the safety measure in place, the actual daily gain/loss is represented
by random variable Y:
-c X <-c,
Y =
X
X >-c
Find the PDF of Y. (4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad306ca9-2ffb-49a4-9b86-39f2ab9633a1%2Fb729d8e5-4904-47a3-b637-3b66cefe7229%2Flmusbaa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem # 4. Yet another variation on a vaguely familiar theme...:
A startup has developed a questionable AI-based app for automatic trading on the stock
market. The following probabilistic model is used to analyze and predict the daily gain/loss
performance of the app. Let random variable R represent the anticipated range of fluctuation
(gain or loss) in the value of the managed stock portfolio during the day, measured in millions
of dollars, whose PDF is fR(r) = re¯u(r). Let random variable X be the actual gain/loss in
value by the end of the day. It was observed that when R takes the value r, X is distributed
uniformly in the range [—r,r], i.e., it ranges between a loss and a gain of r million dollars.
a) Find the marginal density of the actual gain/loss, fx(x).
b) Are R and X uncorrelated, orthogonal and/or independent?
c) Having observed the actual gain/loss of the portfolio at the end of the day, it is desirable to
update the model for the range R. Find the conditional PDF fRx(r|x).
d) An optional safety measure is offered to avoid excessive losses. It stops further trading for
the day whenever the actual loss reaches a certain limit, namely, a loss of c million
dollars. Thus, with the safety measure in place, the actual daily gain/loss is represented
by random variable Y:
-c X <-c,
Y =
X
X >-c
Find the PDF of Y. (4)
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