(a) Let X be the number of sensors that are activated by a nearby motion. What is the distribution of X? Define the values X can take, type and parameters of its distribution, and its probability mass function. (b) Compute the probability that the motion detector in fact detects a motion.
(a) Let X be the number of sensors that are activated by a nearby motion. What is the distribution of X? Define the values X can take, type and parameters of its distribution, and its probability mass function. (b) Compute the probability that the motion detector in fact detects a motion.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Please answer correctly. Write by hand please and make sure you write the formula you use. Attached is the formula sheet use this please.
![Axioms of Probability
Also Note:
1. P(S)=1
For any two events A and B,
2. For any event E, 0< P(E) < 1
P(A) = P(An B) + P(AnB')
3. For any two mutually exclusive events,
and
P(EUF) = P(E) + P(F)
P(AnB) = P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E) + P(F) – P(ENF)
P(A|B) - P(A)
or
Conditional Probability
P(AN B) = P(A)P(B).
P(B|A) =
P(ANB)
P(A)
Bayes' Theorem:
Total Probability Rule
P(AB)P(B)
P(A|B)P(B) + P(A|B')P(B")
P(A) = P(A|B)P(B) + P(A|B')P(B')
P(B|A) =
Similarly.
Similarly,
P(A) -P(A|E1)P(E1) + P(A|E2)P(E2)+
.+ P(A|E)P(E)
P(B|E,)P(E,)
P(B\E,)P(E,)+ P(B\E,)P(E,) + ·…· + P(B\E,)P(E,
P(E\B) =
Probability Mass and Density Functions
Cumulative Distribution Function
• F(z) = P(X S)
If X is a discrete r.v:
lim,- F(r) = 0
P(X = r) = S(x)
Es(z) = 1 (total probability)
• lim, F(x) -1
• F(x) = Eys/(r) if X is a discrete r.v
• P(a < X < b) =- F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• Elh(X)] =E, h(x)f(x) if X is a discrete r.v.
• E'h(X)] = [ h(r)f(z)dr if X is a continu-
• E[X] =E, zf(r) if X is a discrete r.v.
• E[X] = S, ={(z)dz if X is a contimuous r.v.
ous r.v.
• Var(X) - E[X] - E[XP
• E'aX + b] = aE[X] + b
• Var(aX + b) = a³Var(X)
• Var(X) = E|(X - E|X])"
Common Discrete Distributions
•X- Bernoulli(p).
if z = 1;
1-p ifz-0
f(2) =
E[X] = P, Var(X) = p(1 – p).
• X - Geometric(p),
S(2) = (1 – p)*-'p, x € {1,2,.., E[X] = }, Var(X) = .
Geometrie Series: E - for 0 <q<1
%3D
• X- Binomial(n, p).
S(z) = (")(1– p)"-Y,IE {0,1,..., n}, E[X] = np, Var(X) = np(1 – p).
• X- Negative Binomial(r, p).
S(2) = ()(1 – p)*-"P,1€ {r,r+1,..} E[X] = 5, Var(X) = .
• X - Hypergeometric(n, M, N),
, EJx] = n \, Var(X) = =N(1- N).
S(z) =
• X- Poisson(M),
S(z) = , z e {0, 1, ..}, E[X] = At, Var(X)= t.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b0a5411-bdb0-4804-8919-e2df25fb5bff%2F17f14173-5df7-4b6d-9530-4466e94f5f19%2Frxedqw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Axioms of Probability
Also Note:
1. P(S)=1
For any two events A and B,
2. For any event E, 0< P(E) < 1
P(A) = P(An B) + P(AnB')
3. For any two mutually exclusive events,
and
P(EUF) = P(E) + P(F)
P(AnB) = P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E) + P(F) – P(ENF)
P(A|B) - P(A)
or
Conditional Probability
P(AN B) = P(A)P(B).
P(B|A) =
P(ANB)
P(A)
Bayes' Theorem:
Total Probability Rule
P(AB)P(B)
P(A|B)P(B) + P(A|B')P(B")
P(A) = P(A|B)P(B) + P(A|B')P(B')
P(B|A) =
Similarly.
Similarly,
P(A) -P(A|E1)P(E1) + P(A|E2)P(E2)+
.+ P(A|E)P(E)
P(B|E,)P(E,)
P(B\E,)P(E,)+ P(B\E,)P(E,) + ·…· + P(B\E,)P(E,
P(E\B) =
Probability Mass and Density Functions
Cumulative Distribution Function
• F(z) = P(X S)
If X is a discrete r.v:
lim,- F(r) = 0
P(X = r) = S(x)
Es(z) = 1 (total probability)
• lim, F(x) -1
• F(x) = Eys/(r) if X is a discrete r.v
• P(a < X < b) =- F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• Elh(X)] =E, h(x)f(x) if X is a discrete r.v.
• E'h(X)] = [ h(r)f(z)dr if X is a continu-
• E[X] =E, zf(r) if X is a discrete r.v.
• E[X] = S, ={(z)dz if X is a contimuous r.v.
ous r.v.
• Var(X) - E[X] - E[XP
• E'aX + b] = aE[X] + b
• Var(aX + b) = a³Var(X)
• Var(X) = E|(X - E|X])"
Common Discrete Distributions
•X- Bernoulli(p).
if z = 1;
1-p ifz-0
f(2) =
E[X] = P, Var(X) = p(1 – p).
• X - Geometric(p),
S(2) = (1 – p)*-'p, x € {1,2,.., E[X] = }, Var(X) = .
Geometrie Series: E - for 0 <q<1
%3D
• X- Binomial(n, p).
S(z) = (")(1– p)"-Y,IE {0,1,..., n}, E[X] = np, Var(X) = np(1 – p).
• X- Negative Binomial(r, p).
S(2) = ()(1 – p)*-"P,1€ {r,r+1,..} E[X] = 5, Var(X) = .
• X - Hypergeometric(n, M, N),
, EJx] = n \, Var(X) = =N(1- N).
S(z) =
• X- Poisson(M),
S(z) = , z e {0, 1, ..}, E[X] = At, Var(X)= t.
Transcribed Image Text:A motion detector has 3 sensors. Sensors capability of detecting motion is independent of each other.
When an object passes by the detector, each sensor will either be successfully activated with probability
p = or will fail to detect any motion. The object is successfully detected only when the
majority of the sensors are activated.
(a) Let X be the number of sensors that are activated by a nearby motion. What is the distribution
of X? Define the values X can take, type and parameters of its distribution, and its probability
mass function.
(b) Compute the probability that the motion detector in fact detects a motion.
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