4.37 Let X be a continuous random variable with probability density f(x) and finite expected value E(X). (a) What constant c minimizes E[(X- c)2] and what is the minimal age value of E[(X— c)²]?qog voiloq 2 (b) Prove that E(|X— c)) is minimal if c is chosen equal to the median of X.
4.37 Let X be a continuous random variable with probability density f(x) and finite expected value E(X). (a) What constant c minimizes E[(X- c)2] and what is the minimal age value of E[(X— c)²]?qog voiloq 2 (b) Prove that E(|X— c)) is minimal if c is chosen equal to the median of X.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 30CR
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Questions number 4.37
![4 Continuous
4.35 What are the expected value and the standard deviation of the area of
the circle whose radius is a random variable X with density function
f(x) = 1 for 0 < x < 1 and f(x) = 0 otherwise?
4.36 A point Q is chosen at random inside a sphere with radius r. What are
the expected value and the standard deviation of the distance from the
center of the sphere to the point Q?
4.37 Let X be a continuous random variable with probability density f(x) and
finite expected value E(X).
(a) What constant c minimizes E[(X - c)2] and what is the minimal
value of E[(X - c)²]?
com (b) Prove that E(X- c) is minimal if c is chosen equal to the median
gibsour of X.
4.38 Consider again Problem 4.16. Calculate the expected value and the stan-
dard deviation of the height above the ground when the ferris wheel stops.
4.39 In an inventory system, a replenishment order is placed when the stock
on hand of a certain product drops to the level s, where the reorder point
s is a given positive number. The total demand for the product during
the lead time of the replenishment order has the probability density
f(x) de-ix for r
What are th
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Transcribed Image Text:4 Continuous
4.35 What are the expected value and the standard deviation of the area of
the circle whose radius is a random variable X with density function
f(x) = 1 for 0 < x < 1 and f(x) = 0 otherwise?
4.36 A point Q is chosen at random inside a sphere with radius r. What are
the expected value and the standard deviation of the distance from the
center of the sphere to the point Q?
4.37 Let X be a continuous random variable with probability density f(x) and
finite expected value E(X).
(a) What constant c minimizes E[(X - c)2] and what is the minimal
value of E[(X - c)²]?
com (b) Prove that E(X- c) is minimal if c is chosen equal to the median
gibsour of X.
4.38 Consider again Problem 4.16. Calculate the expected value and the stan-
dard deviation of the height above the ground when the ferris wheel stops.
4.39 In an inventory system, a replenishment order is placed when the stock
on hand of a certain product drops to the level s, where the reorder point
s is a given positive number. The total demand for the product during
the lead time of the replenishment order has the probability density
f(x) de-ix for r
What are th
-
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