ECEN4503 Random Signals Week 4 Dr. George Scheets Read 4.2 - 4.4, 4.6 Problems 4.23, 35 – 37, 89 – 91 Quiz #2 Results Hi = 10, Low = 0.7, Ave = 5.80, σ = 3.33 Quiz 2B scores bumped by +1

Examples of Gaussian Noise Radio Static (Thermal Noise) Analog TV "snow" 2 seconds of White Noise

Unconditional PDF X ≡ Clearance P( 0.01 < X < 0.02) = # favorable outcomes # of outcomes = 56/8000 = 0.0070 We can do this for every bin. Result should ∑ = 1.

Conditional PDF Suppose we remove all devices (103 of them) with clearance < 0.01 inches. 7,897 devices remain.

Conditional PDF X ≡ Clearance P( 0.01 < X < 0.02 | X > 0.01) = # favorable outcomes = 56/7897 = 0.0070913 # of outcomes We can do this for every bin. Result should ∑ = 1.

To find a Conditional PDF... Start with unconditional PDF fX(x). Throw out region not needed Normalize remaining area. -∞ +∞ ∫ fX(x)dx = 0.85?

To find a Conditional PDF... Start with unconditional PDF fX(x). Throw out region not needed Normalize remaining area. -∞ +∞ ∫ [fX(x)/0.85]dx = 1.0 [Bracketed term] is the conditional PDF fX(x | ??).

∫ ∑ Expected Values E[ * ] = (*) fX(x) dx -∞ +∞ ∫ E[ * ] = (*) fX(x) dx E[ * ] = (*i)P(*i) (discrete) ∑ all possible outcomes

Standard Deviation Square root of average squared deviation from the mean. Example: X = {6, 8, 10} Average = (6+8+10)/3 = 8 Average |Deviation| from the mean = (2 + 0 + 2)/3 = 4/3 = 1.333 Average squared deviation from the mean = (22 + 0 + 22)/3 = 8/3 = 2.667 Standard Deviation = 2.6670.5 = 1.633

Random Signals Trivia "In the United States, HIV is transmitted from an infected mother to her infant about 30% of the time. At least 6,000 HIV infected women given birth in the United States each year, and therefore a minimum of 1,800 children become infected in a year's time." Science, 17 July 1992

70% "Share of students who consider themselves above average in mathematical ability- a mathematical impossibility." "In Praise of the Ordinary Child" Time Magazine, 3 August 2015

Neb. lottery draws same numbers twice in row Neb. lottery draws same numbers twice in row. The odds of such an odd occurrence? One in a million "Lottery spokesman Brian Rockey says one of two lottery computers that randomly generate combinations picked the numbers 1, 9 and 6 — in that order — for Monday night's drawing. He says the other computer picked the same three numbers Tuesday in the same sequence." -Associated Press, 22 Jan 2009

Let's Make a Deal 3 Doors Behind 2 doors is a decent prize Behind 1 door is a fabulous prize Contestant picks a door (not opened) Host opens a door with a decent prize Does not open door picked by contestant Contestant has option of switching doors Gets whatever is behind picked door

Your Money or Your Life Behind 2 doors Mafia hit man with a shotgun Behind 1 door is $1,000,000,000

Your Money or Your Life 1 2 3 Suppose you pick door number 3

Your Money or Your Life 2 3 From a probability viewpoint, what tactic will maximize your chance of winning? Stay with door originally chosen Change doors Mentally flip a coin. It doesn't matter, the odds are same for all above.

Your Money or Your Life 2 From a probability viewpoint, what tactic will maximize your chance of winning? Stay with door originally chosen Change doors Mentally flip a coin. It doesn't matter, the odds are the same.

Strategy... Stay with door originally chosen Change doors You must pick correct door initially 33% chance of winning. Change doors You win if you initially pick wrong door 67% chance of winning. Mentally flip a coin after 1 door is opened You have a 50% chance of winning. It doesn't matter, the odds are the same It matters.

Lottery Assistance