A shot is fired at a very large circular target. The horizontal and ve

kenziebabyyy4e

kenziebabyyy4e

Answered question

2021-11-25

A shot is fired at a very large circular target. The horizontal and vertical coordinates of the point of impact are independent random variables each having a standard normal density. Here the center of the target is taken as the origin. What is the density function of the distance from the center of the target to the point of impact? What are the expected value and the mode of this distance?

Answer & Explanation

Heack1991

Heack1991

Beginner2021-11-26Added 13 answers

Step 1
Let,
Horizontal co-ordinates be denoted by X.
Vertical co-ordinates be denoted by Y.
Distance between center of target to the point of impact is calculated using Pythagoras Theorem.
Let Z denote the distance.
So, Z=X2+Y2
Since both X and Y follow standard normal distributions, using properties of sampling Z will follow Chi distribution with degrees of freedom 2.
Step 2
Expected value of Z can be calculated by finding out the mean.
Mean=2Γ(k+12)Γ(k2), where k is the degrees of freedom, which is 2.
=2Γ(32)Γ(12)=20.8861.772=0.707
Mode of the distance will be 21=1

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