ECEN4503 Random Signals Lecture #18 24 February 2014 Dr. George Scheets n Read 5.3 & 5.5 n Problems 5.1, 5.4, 5.11 n Exam #1 Friday n Quiz 4 Results Hi = 10.0, Low = 2.3, Average = 8.29 σ = 1.89

ECEN4503 Random Signals Lecture #19 26 February 2014 Dr. George Scheets n Read 5.6 & 5.7 n Problems: 5.8abd, 5.12, 5.16 n Test next time!!

Exam #1 n Covers Chapter 1 – 5.3 (text) n Day #1 - 2nd Order PDF's (notes) n Open books, notes. n Closed instructor, neighbor. n For All Quizzes & Tests... u Calculators (highly recommended) u Smart Phones are NOT allowed

Possible Exam Topics S Reading Chapters HW Lectures Anything in the circles is fair game.

Budget Your Time!!!!!!!!!!! n 4 page exam, 25 points per page n Instructor not available for hints n Typically u 1 page solvable < minutes u 2 pages solvable in minutes u 1 page to separate Men from Boys & Girls from Women n Recommendation u 1st pass: Spend < minutes/page u 2nd pass: Go to problem(s) you know best

2nd Order PDF's n Volume should equal 1 n To get marginal PDF f X (x) or f Y (Y) u Integrate out variable you don't want f XY (x,y) dx dy = 1 f XY (x,y) dy = f X (x)

3D Histogram Probabilities will sum to Age Weight 11 f AW (a,w) dw = f A (a)f AW (21,w) dw = f A (21)

Marginal PDF - Age 30/85 20/85 10/85 Probability

Age & Middle Finger Length S.I.? Linear Regression Length = *Age If S.I., slope = 0

S.I. Random Variables n P(X∩Y) = P(X)P(Y) n f XY (x,y) = f X (x) f Y (y) n E[XY] =E[X]E[Y] u ∫∫xy f XY (x,y)dxdy

Sum of S.I. Random Variables n Given a noise voltage N... Time Volts 0

Bin Count Time Volts

...With Uniform Voltage PDF... n volts f N (n) 0+1 1/2 Suppose you add the number 6 to this. What's the output PDF of N+6?

Time domain signal is now... Time Volts 6

PDF of Y = N + 6 y volts f Y (y) /2

f Y (y) = f N (n) f 6 (n) n volts f N (n) 0+1 1/2 n volts f 6 (n) +6 1 Convolution of an arbitrary function with a delta function Shifts and centers function's origin to delta function's location.

Add together two die rolls x dots f die roll #1 (x) /6 x dots f die roll #2 (x) /6 Shift & center origin of arbitrary function (delta functions on left) to location of delta functions on the right. Multiply. Repeat 6 times. Add overlaps.

Add together two S.I. die rolls y dots f sum (y) /36 4/36 2/36 1/36 3/36 5/36

Matlab Convolution Tool n This code can help you understand convolution. Available near the bottom of the link below. n This code can help you understand convolution. Available near the bottom of the link below.