Lecture Notes 1 Probability and Random Variables • Probability Spaces • Conditional Probability and Independence • Random Variables • Functions of a Random Variable • Generation of a Random Variable • Jointly Distributed Random Variables • Scalar detection EE 278B: Probability and Random Variables 1–1 Probability Theory • Probability theory provides the mathematical rules for assigning probabilities to outcomes of random experiments, e.g., coin flips, packet arrivals, noise voltage • Basic elements of probability theory: ◦ Sample space Ω: set of all possible “elementary” or “finest grain” outcomes of the random experiment ◦ Set of events F : set of (all?) subsets of Ω — an event A ⊂ Ω occurs if the outcome ω ∈ A ◦ Probability measure P: function over F that assigns probabilities to events according to the axioms of probability (see below) • Formally, a probability space is the triple (Ω, F , P) EE 278B: Probability and Random Variables 1–2
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Lecture Notes 1
Probability and Random Variables
• Probability Spaces
• Conditional Probability and Independence
• Random Variables
• Functions of a Random Variable
• Generation of a Random Variable
• Jointly Distributed Random Variables
• Scalar detection
EE 278B: Probability and Random Variables 1 – 1
Probability Theory
• Probability theory provides the mathematical rules for assigning probabilities tooutcomes of random experiments, e.g., coin flips, packet arrivals, noise voltage
• Basic elements of probability theory:
Sample space Ω: set of all possible “elementary” or “finest grain” outcomesof the random experiment
Set of events F : set of (all?) subsets of Ω—an event A ⊂ Ω occurs if theoutcome ω ∈ A
Probability measure P: function over F that assigns probabilities to eventsaccording to the axioms of probability (see below)
• Formally, a probability space is the triple (Ω,F ,P)
EE 278B: Probability and Random Variables 1 – 2
Axioms of Probability
• A probability measure P satisfies the following axioms:
1. P(A) ≥ 0 for every event A in F2. P(Ω) = 1
3. If A1, A2, . . . are disjoint events—i.e., Ai ∩Aj = ∅, for all i 6= j —then
P
( ∞⋃
i=1
Ai
)
=
∞∑
i=1
P(Ai)
• Notes:
P is a measure in the same sense as mass, length, area, and volume—allsatisfy axioms 1 and 3
Unlike these other measures, P is bounded by 1 (axiom 2)
This analogy provides some intuition but is not sufficient to fully understandprobability theory—other aspects such as conditioning and independence areunique to probability
EE 278B: Probability and Random Variables 1 – 3
Discrete Probability Spaces
• A sample space Ω is said to be discrete if it is countable
• Examples:
Rolling a die: Ω = 1, 2, 3, 4, 5, 6 Flipping a coin n times: Ω = H, Tn, sequences of heads/tails of length n
Flipping a coin until the first heads occurs: Ω = H,TH, TTH, TTTH, . . .
• For discrete sample spaces, the set of events F can be taken to be the set of allsubsets of Ω, sometimes called the power set of Ω
• Example: For the coin flipping experiment,
F = ∅, H, T, Ω
• F does not have to be the entire power set (more on this later)
EE 278B: Probability and Random Variables 1 – 4
• The probability measure P can be defined by assigning probabilities to individualoutcomes— single outcome events ω—so that:
P(ω) ≥ 0 for every ω ∈ Ω
∑
ω∈Ω
P(ω) = 1
• The probability of any other event A is simply
P(A) =∑
ω∈A
P(ω)
• Example: For the die rolling experiment, assign
P(i) = 16 for i = 1, 2, . . . , 6
The probability of the event “the outcome is even,” A = 2, 4, 6, isP(A) = P(2) + P(4) + P(6) = 3
6 = 12
EE 278B: Probability and Random Variables 1 – 5
Continuous Probability Spaces
• A continuous sample space Ω has an uncountable number of elements
• Examples:
Random number between 0 and 1: Ω = (0, 1 ]
Point in the unit disk: Ω = (x, y) : x2 + y2 ≤ 1 Arrival times of n packets: Ω = (0,∞)n
• For continuous Ω, we cannot in general define the probability measure P by firstassigning probabilities to outcomes
• To see why, consider assigning a uniform probability measure over (0, 1 ]
In this case the probability of each single outcome event is zero
How do we find the probability of an event such as A = [0.25, 0.75]?
EE 278B: Probability and Random Variables 1 – 6
• Another difference for continuous Ω: we cannot take the set of events F as thepower set of Ω. (To learn why you need to study measure theory, which isbeyond the scope of this course)
• The set of events F cannot be an arbitrary collection of subsets of Ω. It mustmake sense, e.g., if A is an event, then its complement Ac must also be anevent, the union of two events must be an event, and so on
• Formally, F must be a sigma algebra (σ-algebra, σ-field), which satisfies thefollowing axioms:
1. ∅ ∈ F2. If A ∈ F then Ac ∈ F3. If A1, A2, . . . ∈ F then
⋃∞i=1Ai ∈ F
• Of course, the power set is a sigma algebra. But we can define smallerσ-algebras. For example, for rolling a die, we could define the set of events as
F = ∅, odd, even, Ω
EE 278B: Probability and Random Variables 1 – 7
• For Ω = R = (−∞,∞) (or (0,∞), (0, 1), etc.) F is typically defined as thefamily of sets obtained by starting from the intervals and taking countableunions, intersections, and complements
• The resulting F is called the Borel field
• Note: Amazingly there are subsets in R that cannot be generated in this way!(Not ones that you are likely to encounter in your life as an engineer or even asa mathematician)
• To define a probability measure over a Borel field, we first assign probabilities tothe intervals in a consistent way, i.e., in a way that satisfies the axioms ofprobability
For example to define uniform probability measure over (0, 1), we first assignP((a, b)) = b− a to all intervals
• In EE 278 we do not deal with sigma fields or the Borel field beyond (kind of)knowing what they are
EE 278B: Probability and Random Variables 1 – 8
Useful Probability Laws
• Union of Events Bound :
P(
n⋃
i=1
Ai
)
≤n∑
i=1
P(Ai)
• Law of Total Probability : Let A1, A2, A3, . . . be events that partition Ω, i.e.,disjoint (Ai ∩Aj = ∅ for i 6= j) and
⋃
iAi = Ω. Then for any event B
P(B) =∑
iP(Ai ∩B)
The Law of Total Probability is very useful for finding probabilities of sets
EE 278B: Probability and Random Variables 1 – 9
Conditional Probability
• Let B be an event such that P(B) 6= 0. The conditional probability of event Agiven B is defined to be
P(A |B) =P(A ∩B)
P(B)=
P(A,B)
P(B)
• The function P(· |B) is a probability measure over F , i.e., it satisfies theaxioms of probability
• Chain rule: P(A,B) = P(A)P(B |A) = P(B)P(A |B) (this can be generalizedto n events)
• The probability of event A given B , a nonzero probability event— thea posteriori probability of A—is related to the unconditional probability ofA—the a priori probability—by
P(A |B) =P(B |A)P(B)
P(A)
This follows directly from the definition of conditional probability
EE 278B: Probability and Random Variables 1 – 10
Bayes Rule
• Let A1, A2, . . . , An be nonzero probability events that partition Ω, and let B bea nonzero probability event
• We know P(Ai) and P(B |Ai), i = 1, 2, . . . , n, and want to find the a posterioriprobabilities P(Aj |B), j = 1, 2, . . . , n
• We know that
P(Aj |B) =P(B |Aj)
P(B)P(Aj)
• By the law of total probability
P(B) =
n∑
i=1
P(Ai, B) =
n∑
i=1
P(Ai)P(B |Ai)
• Substituting, we obtain Bayes rule
P(Aj |B) =P(B |Aj)
∑ni=1P(Ai)P(B |Ai)
P(Aj), j = 1, 2, . . . , n
• Bayes rule also applies to a (countably) infinite number of events
EE 278B: Probability and Random Variables 1 – 11
Independence
• Two events are said to be statistically independent if
P(A,B) = P(A)P(B)
• When P(B) 6= 0, this is equivalent to
P(A |B) = P(A)
In other words, knowing whether B occurs does not change the probability of A
• The events A1, A2, . . . , An are said to be independent if for every subsetAi1, Ai2, . . . , Aik of the events,
P(Ai1, Ai2, . . . , Aik) =k∏
j=1
P(Aij)
• Note: P(A1, A2, . . . , An) =∏n
j=1P(Ai) is not sufficient for independence
EE 278B: Probability and Random Variables 1 – 12
Random Variables
• A random variable (r.v.) is a real-valued function X(ω) over a sample space Ω,i.e., X : Ω → R
Ω
ω X(ω)
• Notations:
We use upper case letters for random variables: X, Y, Z, Φ, Θ, . . .
We use lower case letters for values of random variables: X = x means thatrandom variable X takes on the value x, i.e., X(ω) = x where ω is theoutcome
EE 278B: Probability and Random Variables 1 – 13
Specifying a Random Variable
• Specifying a random variable means being able to determine the probability thatX ∈ A for any Borel set A ⊂ R, in particular, for any interval (a, b ]
• To do so, consider the inverse image of A under X , i.e., ω : X(ω) ∈ A
R
set A
inverse image of A under X(ω), i.e., ω : X(ω) ∈ A
• Since X ∈ A iff ω ∈ ω : X(ω) ∈ A,P(X ∈ A) = P(ω : X(ω) ∈ A) = Pω : X(ω) ∈ A
Shorthand: P(set description) = Pset description
EE 278B: Probability and Random Variables 1 – 14
Cumulative Distribution Function (CDF)
• We need to be able to determine PX ∈ A for any Borel set A ⊂ R, i.e., anyset generated by starting from intervals and taking countable unions,intersections, and complements
• Hence, it suffices to specify PX ∈ (a, b ] for all intervals. The probability ofany other Borel set can be determined by the axioms of probability
• Equivalently, it suffices to specify its cumulative distribution function (cdf):
FX(x) = PX ≤ x = PX ∈ (−∞, x ] , x ∈ R
• Properties of cdf:
FX(x) ≥ 0
FX(x) is monotonically nondecreasing, i.e., if a > b then FX(a) ≥ FX(b)
FX(x)
1
x
EE 278B: Probability and Random Variables 1 – 15
Limits: limx→+∞
FX(x) = 1 and limx→−∞
FX(x) = 0
FX(x) is right continuous, i.e., FX(a+) = limx→a+ FX(x) = FX(a)
PX = a = FX(a)− FX(a−), where FX(a−) = limx→a− FX(x)
For any Borel set A, PX ∈ A can be determined from FX(x)
• Notation: X ∼ FX(x) means that X has cdf FX(x)
EE 278B: Probability and Random Variables 1 – 16
Probability Mass Function (PMF)
• A random variable is said to be discrete if FX(x) consists only of steps over acountable set X
PSfrag
1
x
• Hence, a discrete random variable can be completely specified by the probability
mass function (pmf)
pX(x) = PX = x for every x ∈ XClearly pX(x) ≥ 0 and
∑
x∈X pX(x) = 1
• Notation: We use X ∼ pX(x) or simply X ∼ p(x) to mean that the discreterandom variable X has pmf pX(x) or p(x)
EE 278B: Probability and Random Variables 1 – 17
• Famous discrete random variables:
Bernoulli : X ∼ Bern(p) for 0 ≤ p ≤ 1 has the pmf
pX(1) = p and pX(0) = 1− p
Geometric : X ∼ Geom(p) for 0 ≤ p ≤ 1 has the pmf
pX(k) = p(1− p)k−1 , k = 1, 2, 3, . . .
Binomial : X ∼ Binom(n, p) for integer n > 0 and 0 ≤ p ≤ 1 has the pmf
pX(k) =
(
n
k
)
pk(1− p)n−k, k = 0, 1, 2, . . .
Poisson: X ∼ Poisson(λ) for λ > 0 has the pmf
pX(k) =λk
k!e−λ , k = 0, 1, 2, . . .
Remark: Poisson is the limit of Binomial for np = λ as n → ∞, i.e., for everyk = 0, 1, 2, . . ., the Binom(n, λ/n) pmf
pX(k) → λk
k!e−λ as n → ∞
EE 278B: Probability and Random Variables 1 – 18
Probability Density Function (PDF)
• A random variable is said to be continuous if its cdf is a continuous function
PSfrag
1
x
• If FX(x) is continuous and differentiable (except possibly over a countable set),then X can be completely specified by a probability density function (pdf)fX(x) such that
FX(x) =
∫ x
−∞fX(u) du
• If FX(x) is differentiable everywhere, then (by definition of derivative)
fX(x) =dFX(x)
dx
= lim∆x→0
F (x + ∆x) − F (x)
∆x= lim
∆x→0
Px < X ≤ x + ∆x∆x
EE 278B: Probability and Random Variables 1 – 19
• Properties of pdf:
fX(x) ≥ 0
∫ ∞
−∞fX(x) dx = 1
For any event (Borel set) A ⊂ R,
PX ∈ A =
∫
x∈A
fX(x) dx
In particular,
Px1 < X ≤ x2 =
∫ x2
x1
fX(x) dx
• Important note: fX(x) should not be interpreted as the probability that X = x.In fact, fX(x) is not a probability measure since it can be > 1
• Notation: X ∼ fX(x) means that X has pdf fX(x)
EE 278B: Probability and Random Variables 1 – 20
• Famous continuous random variables:
Uniform: X ∼ U[a, b ] where a < b has pdf
fX(x) =
1b−a if a ≤ x ≤ b
0 otherwise
Exponential : X ∼ Exp(λ) where λ > 0 has pdf
fX(x) =
λe−λx if x ≥ 0
0 otherwise
Laplace: X ∼ Laplace(λ) where λ > 0 has pdf
fX(x) =1
2λe−λ|x|
Gaussian: X ∼ N (µ, σ2) with parameters µ (the mean) and σ2 (thevariance, σ is the standard deviation) has pdf
fX(x) =1√2πσ2
e−(x−µ)2
2σ2
EE 278B: Probability and Random Variables 1 – 21
The cdf of the standard normal random variable N (0, 1) is
Φ(x) =
∫ x
−∞
1√2π
e−u2
2 du
Define the function Q(x) = 1− Φ(x) = PX > xN (0, 1)
x
Q(x)
The Q(·) function is used to compute PX > a for any Gaussian r.v. X :
Given Y ∼ N (µ, σ2), we represent it using the standard X ∼ N (0, 1) as
Y = σX + µ
Then
PY > y = P
X >y − µ
σ
= Q
(
y − µ
σ
)
The complementary error function is erfc(x) = 2Q(√2x)
EE 278B: Probability and Random Variables 1 – 22
Functions of a Random Variable
• Suppose we are given a r.v. X with known cdf FX(x) and a function y = g(x).What is the cdf of the random variable Y = g(X)?
• We useFY (y) = PY ≤ y = Px : g(x) ≤ y
x y
y
x : g(x) ≤ y
EE 278B: Probability and Random Variables 1 – 23
• Example: Quadratic function. Let X ∼ FX(x) and Y = X2. We wish to findFY (y)
y
x
y
√y−√
y
If y < 0, then clearly FY (y) = 0. Consider y ≥ 0,
FY (y) = P −√y < X ≤ √
y = FX (√y)− FX (−√
y )
If X is continuous with density fX(x), then
fY (y) =1
2√y
(
fX(+√y) + fX(−√
y))
EE 278B: Probability and Random Variables 1 – 24
• Remark: In general, let X ∼ fX(x) and Y = g(X) be differentiable. Then
fY (y) =k∑
i=1
fX(xi)
|g′(xi)|,
where x1, x2, . . . are the solutions of the equation y = g(x) and g′(xi) is thederivative of g evaluated at xi
EE 278B: Probability and Random Variables 1 – 25
• Example: Limiter. Let X ∼ Laplace(1), i.e., fX(x) = (1/2)e−|x|, and let Y bedefined by the function of X shown in the figure. Find the cdf of Y
y
x+1
−1
+a
−a
To find the cdf of Y , we consider the following cases
y < −a: Here clearly FY (y) = 0
y = −a: Here
FY (−a) = FX(−1)
=
∫ −1
−∞12e
x dx = 12e
−1
EE 278B: Probability and Random Variables 1 – 26
−a < y < a: Here
FY (y) = PY ≤ y= P aX ≤ y
= P
X ≤ y
a
= FX
(y
a
)
= 12e
−1 +
∫ y/a
−1
12e
−|x| dx
y ≥ a: Here FY (y) = 1
Combining the results, the following is a sketch of the cdf of Y
FY (y)
ya−a
EE 278B: Probability and Random Variables 1 – 27
Generation of Random Variables
• Generating a r.v. with a prescribed distribution is often needed for performingsimulations involving random phenomena, e.g., noise or random arrivals
• First let X ∼ F (x) where the cdf F (x) is continuous and strictly increasing.Define Y = F (X), a real-valued random variable that is a function of X
What is the cdf of Y ?
Clearly, FY (y) = 0 for y < 0, and FY (y) = 1 for y > 1
For 0 ≤ y ≤ 1, note that by assumption F has an inverse F−1, so
FY (y) = PY ≤ y = PF (X) ≤ y = PX ≤ F−1(y) = F (F−1(y)) = y
Thus Y ∼ U [ 0, 1 ], i.e., Y is a uniformly distributed random variable
• Note: F (x) does not need to be invertible. If F (x) = a is constant over someinterval, then the probability that X lies in this interval is zero. Without loss ofgenerality, we can take F−1(a) to be the leftmost point of the interval
• Conclusion: We can generate a U[ 0, 1 ] r.v. from any continuous r.v.
EE 278B: Probability and Random Variables 1 – 28
• Now, let’s consider the more useful scenario where we are given X ∼ U[ 0, 1 ] (arandom number generator) and wish to generate a random variable Y withprescribed cdf F (y), e.g., Gaussian or exponential
x = F (y)
y = F−1(x)
1
• If F is continuous and strictly increasing, set Y = F−1(X). To show Y ∼ F (y),
FY (y) = PY ≤ y= PF−1(X) ≤ y= PX ≤ F (y)= F (y) ,
since X ∼ U[ 0, 1 ] and 0 ≤ F (y) ≤ 1
EE 278B: Probability and Random Variables 1 – 29
• Example: To generate Y ∼ Exp(λ), set
Y = −1
λln(1−X)
• Note: F does not need to be continuous for the above to work. For example, togenerate Y ∼ Bern(p), we set
Y =
0 X ≤ 1− p
1 otherwise
x = F (y)
y
1−p
10
• Conclusion: We can generate a r.v. with any desired distribution from a U[0, 1] r.v.
EE 278B: Probability and Random Variables 1 – 30
Jointly Distributed Random Variables
• A pair of random variables defined over the same probability space are specifiedby their joint cdf
FX,Y (x, y) = PX ≤ x, Y ≤ y , x, y ∈ R
FX,Y (x, y) is the probability of the shaded region of R2
x
y
(x, y)
EE 278B: Probability and Random Variables 1 – 31
• Properties of the cdf:
FX,Y (x, y) ≥ 0
If x1 ≤ x2 and y1 ≤ y2 then FX,Y (x1, y1) ≤ FX,Y (x2, y2)
limy→−∞
FX,Y (x, y) = 0 and limx→−∞
FX,Y (x, y) = 0
limy→∞
FX,Y (x, y) = FX(x) and limx→∞
FX,Y (x, y) = FY (y)
FX(x) and FY (y) are the marginal cdfs of X and Y
limx,y→∞
FX,Y (x, y) = 1
• X and Y are independent if for every x and y
FX,Y (x, y) = FX(x)FY (y)
EE 278B: Probability and Random Variables 1 – 32
Joint, Marginal, and Conditional PMFs
• Let X and Y be discrete random variables on the same probability space
• They are completely specified by their joint pmf :
pX,Y (x, y) = PX = x, Y = y , x ∈ X , y ∈ Y
By axioms of probability,∑
x∈X
∑
y∈YpX,Y (x, y) = 1
• To find pX(x), the marginal pmf of X , we use the law of total probability
pX(x) =∑
y∈Yp(x, y) , x ∈ X
• The conditional pmf of X given Y = y is defined as
• Independence: X and Y are said to be independent if for every (x, y) ∈ X × Y ,
pX,Y (x, y) = pX(x)pY (y) ,
which is equivalent to pX|Y (x|y) = pX(x) for every x ∈ X and y ∈ Y suchthat pY (y) 6= 0
EE 278B: Probability and Random Variables 1 – 34
Joint, Marginal, and Conditional PDF
• X and Y are jointly continuous random variables if their joint cdf is continuousin both x and y
In this case, we can define their joint pdf, provided that it exists, as the functionfX,Y (x, y) such that
FX,Y (x, y) =
∫ x
−∞
∫ y
−∞fX,Y (u, v) du dv , x, y ∈ R
• If FX,Y (x, y) is differentiable in x and y, then
fX,Y (x, y) =∂2F (x, y)
∂x∂y= lim
∆x,∆y→0
Px < X ≤ x+∆x, y < Y ≤ y +∆y∆x∆y
• Properties of fX,Y (x, y):
fX,Y (x, y) ≥ 0
∫ ∞
−∞
∫ ∞
−∞fX,Y (x, y) dx dy = 1
EE 278B: Probability and Random Variables 1 – 35
• The marginal pdf of X can be obtained from the joint pdf via the law of totalprobability:
fX(x) =
∫ ∞
−∞fX,Y (x, y) dy
• X and Y are independent iff fX,Y (x, y) = fX(x)fY (y) for every x, y
• Conditional cdf and pdf: Let X and Y be continuous random variables withjoint pdf fX,Y (x, y). We wish to define FY |X(y |X = x) = PY ≤ y |X = xWe cannot define the above conditional probability as
PY ≤ y, X = xPX = x
because both numerator and denominator are equal to zero. Instead, we defineconditional probability for continuous random variables as a limit
FY |X(y|x) = lim∆x→0
PY ≤ y |x < X ≤ x+∆x
= lim∆x→0
PY ≤ y, x < X ≤ x+∆xPx < X ≤ x+∆x
= lim∆x→0
∫ y
−∞ fX,Y (x, u) du∆x
fX(x)∆x=
∫ y
−∞
fX,Y (x, u)
fX(x)du
EE 278B: Probability and Random Variables 1 – 36
• We then define the conditional pdf in the usual way as
fY |X(y|x) = fX,Y (x, y)
fX(x)if fX(x) 6= 0
• Thus
FY |X(y|x) =∫ y
−∞fY |X(u|x) du
which shows that fY |X(y|x) is a pdf for Y given X = x, i.e.,
Y | X = x ∼ fY |X(y|x)
• independence: X and Y are independent if fX,Y (x, y) = fX(x)fY (y) for every(x, y)
EE 278B: Probability and Random Variables 1 – 37
Mixed Random Variables
• Let Θ be a discrete random variable with pmf pΘ(θ)
• For each Θ = θ with pΘ(θ) 6= 0, let Y be a continuous random variable, i.e.,FY |Θ(y|θ) is continuous for all θ. We define fY |Θ(y|θ) in the usual way
• The conditional pmf of Θ given y can be defined as a limit
pΘ|Y (θ | y) = lim∆y→0
PΘ = θ, y < Y ≤ y +∆yPy < Y ≤ y +∆y
= lim∆y→0
pΘ(θ)fY |Θ(y|θ)∆y
fY (y)∆y=
fY |Θ(y|θ)pΘ(θ)fY (y)
EE 278B: Probability and Random Variables 1 – 38
Bayes Rule for Random Variables
• Bayes Rule for pmfs: Given pX(x) and pY |X(y|x), then
pX|Y (x|y) =pY |X(y|x)
∑
x′∈XpY |X(y|x′)pX(x′)
pX(x)
• Bayes rule for densities: Given fX(x) and fY |X(y|x), then
fX|Y (x|y) =fY |X(y|x)
∫∞−∞ fX(u)fY |X(y|u) du fX(x)
• Bayes rule for mixed r.v.s: Given pΘ(θ) and fY |Θ(y|θ), then
pΘ|Y (θ | y) =fY |Θ(y|θ)
∑
θ′ pΘ(θ′)fY |Θ(y|θ′)
pΘ(θ)
Conversely, given fY (y) and pΘ|Y (θ|y), then
fY |Θ(y|θ) =pΘ|Y (θ|y)
∫
fY (y′)pΘ|Y (θ|y′)dy′fY (y)
EE 278B: Probability and Random Variables 1 – 39
• Example: Additive Gaussian Noise Channel
Consider the following communication channel:
Θ Y
Z ∼ N (0, N)
The signal transmitted is a binary random variable Θ:
Θ =
+1 with probability p
−1 with probability 1− p
The received signal, also called the observation, is Y = Θ+ Z , where Θ and Zare independent
Given Y = y is received (observed), find pΘ|Y (θ|y), the a posteriori pmf of Θ
EE 278B: Probability and Random Variables 1 – 40
Solution: We use Bayes rule
pΘ|Y (θ|y) =fY |Θ(y|θ)
∑
θ′ pΘ(θ′)fY |Θ(y|θ′)
pΘ(θ)
We are given pΘ(θ):
pΘ(+1) = p and pΘ(−1) = 1− p
and fY |Θ(y|θ) = fZ(y − θ):
Y | Θ = +1 ∼ N (+1, N) and Y | Θ = −1 ∼ N (−1, N)
Therefore
pΘ|Y (1|y) =p√2πN
e−(y−1)2
2N
p√2πN
e−(y−1)2
2N +(1 − p)√
2πNe−
(y+1)2
2N
=pe
yN
peyN + (1− p)e−
yN
for −∞ < y < ∞
EE 278B: Probability and Random Variables 1 – 41
Scalar Detection
• Consider the following general digital communication system
Θ ∈ θ0, θ1 Θ(Y ) ∈ θ0, θ1Ynoisychannel decoder
fY |Θ(y|θ)where the signal sent is
Θ =
θ0 with probability p
θ1 with probability 1− p
and the observation (received signal) is
Y | Θ = θ ∼ fY |Θ(y | θ) , θ ∈ θ0, θ1
• We wish to find the estimate Θ(Y ) (i.e., design the decoder) that minimizes theprobability of error :