magine you are measuring heights in the population, bounded between |a, b]. You obtain n iid neasurements X1, X2, . , X, from the population, where each X¡ is a bounded random variable i vith mean E[X;] = µb. We explicitly label the mean for these bounded RVs as µb, instead of µ, to m nore clear we are referencing the mean for these particular RVs rather than just a generic mean. Let -Σι Χ. o get confidence intervals we used concentration inequalities. We can use a concentration inequality pecifically for bounded variables (Hoeffding's inequality), to get that P(|X – E[X]| > e) <2 exp(-) (6-a)? Recall that exp(a) is the exponential function: e“. You can use log to mean the natural log, which is the nverse of the exponential.
magine you are measuring heights in the population, bounded between |a, b]. You obtain n iid neasurements X1, X2, . , X, from the population, where each X¡ is a bounded random variable i vith mean E[X;] = µb. We explicitly label the mean for these bounded RVs as µb, instead of µ, to m nore clear we are referencing the mean for these particular RVs rather than just a generic mean. Let -Σι Χ. o get confidence intervals we used concentration inequalities. We can use a concentration inequality pecifically for bounded variables (Hoeffding's inequality), to get that P(|X – E[X]| > e) <2 exp(-) (6-a)? Recall that exp(a) is the exponential function: e“. You can use log to mean the natural log, which is the nverse of the exponential.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![Imagine you are measuring heights in the population, bounded between [a, b]. You obtain n iid
measurements X1, X2, ... , X, from the population, where each X; is a bounded random variable in [a, b]
with mean E[X;] = µp. We explicitly label the mean for these bounded RVs as µb, instead of µ, to make it
more clear we are referencing the mean for these particular RVs rather than just a generic mean. Let
X = E,Xi-
i=1
To get confidence intervals we used concentration inequalities. We can use a concentration inequality
specifically for bounded variables (Hoeffding's inequality), to get that
P(|X – E[X]| > e) < 2 exp(-2)
(b-a)ª
2ne?
Recall that exp(a) is the exponential function: e". You can use log to mean the natural log, which is the
inverse of the exponential.
You can use the following notation to write these mathematical terms:
n, Xbar, X_i, x, E[X], E[X_i], mu_b, epsilon, delta, sum_{i = 1:n}, exp, log, epsilon^2.
Part a
Express E[X] in terms of the values above, namely µb, a, b, and n. Your final expression should not have a
sum over i. Note: the final expression does not necessarily need to include all variables µb, a, b and n. Show
your steps.
Part b
Derive a 95% confidence interval for E[X], using the above inequality. You can leave the expression in terms
of generic
confidence interval for a = 1, b = 3, n =
ies for a, b and n. If you are uncomfortable leaving it generic, then you can provide the
20. Show your steps.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36a7a892-d2a2-4a6f-a8a2-55ad3cac8d8b%2F0e80554f-01c7-430b-8b74-b5590e26c346%2F3ysttr5_processed.png&w=3840&q=75)
Transcribed Image Text:Imagine you are measuring heights in the population, bounded between [a, b]. You obtain n iid
measurements X1, X2, ... , X, from the population, where each X; is a bounded random variable in [a, b]
with mean E[X;] = µp. We explicitly label the mean for these bounded RVs as µb, instead of µ, to make it
more clear we are referencing the mean for these particular RVs rather than just a generic mean. Let
X = E,Xi-
i=1
To get confidence intervals we used concentration inequalities. We can use a concentration inequality
specifically for bounded variables (Hoeffding's inequality), to get that
P(|X – E[X]| > e) < 2 exp(-2)
(b-a)ª
2ne?
Recall that exp(a) is the exponential function: e". You can use log to mean the natural log, which is the
inverse of the exponential.
You can use the following notation to write these mathematical terms:
n, Xbar, X_i, x, E[X], E[X_i], mu_b, epsilon, delta, sum_{i = 1:n}, exp, log, epsilon^2.
Part a
Express E[X] in terms of the values above, namely µb, a, b, and n. Your final expression should not have a
sum over i. Note: the final expression does not necessarily need to include all variables µb, a, b and n. Show
your steps.
Part b
Derive a 95% confidence interval for E[X], using the above inequality. You can leave the expression in terms
of generic
confidence interval for a = 1, b = 3, n =
ies for a, b and n. If you are uncomfortable leaving it generic, then you can provide the
20. Show your steps.
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