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ECE 341 - Probability and Random Processes for Engineers Homework #8 (write answers here, show work on paper stapled to this cover sheet)
Problem 1.
Write one line of Matlab code that generates array X to contain one thousand
samples of a continuous uniformly-distributed random variable where μ X =− 5
and E [ X^2 ]= 10 :
Problem 2.
Write one line of Matlab code that generates array Y to contain one million
samples of a Gaussian random variable where μ Y =− 2 and Var [ Y ]= 2 :
Problem 3.
Given the following samples of Gaussian random variable X :
-11.6458, 0.4269, -5.3993, 1.1400, 1.8937, 1.2714, 4.
a) Find an unbiased estimate of E [ X ]:
b) Find an unbiased estimate of Var [ X ]:
c) Now assume that μ (^) X =− 3. Find an unbiased estimate of Var [ X ]:
d) Assuming that μ (^) X is unknown but that σ (^) X = 6 , what is the 95% confidence
interval for the answer that you gave in (a)?
ECE 341 - Probability and Random Processes for Engineers Homework #8 Solutions
Problem 1.
Write one line of Matlab code that generates array X to contain one thousand
samples of a continuous uniformly-distributed random variable where μ X =− 3
and E [ X^2 ]= 10 :
To find the desired variance: Var [ X ]= E [ X^2 ]− μ (^2) X = 10 − 9 = 1. Note that for a
continuous uniform r.v., variance = (span) 2 /12 so here the span must equal 12.
The range of X is therefore − 3 − 212 ≤ X ≤− 3 + 212 :
X = sqrt(12)(rand(1,1000)-0.5) - 3;*
Problem 2.
Write one line of Matlab code that generates array Y to contain one million
samples of a Gaussian random variable where μ Y^ =− 2 and Var [ Y ]= 2 :
Var [ Y ]= 2 so then σ (^) Y = 2 :
Y = sqrt(2)randn(1,1000000) - 2;*
Problem 3.
Given the following samples of Gaussian random variable X :
-11.6458, 0.4269, -5.3993, 1.1400, 1.8937, 1.2714, 4.
a) Find an unbiased estimate of E [ X ]:
=
n
i
Mn X n xi 1
b) Find an unbiased estimate of Var [ X ]:
=
n
i n xi^ Mn X 1
( ( ))^2
