a) What is the probability that it takes more than 10 minutes for the volunteer to give out the first flyer? b) What is the expected number of flyers the volunteer would distribute in 6 hours? c) Suppose the volunteer has 3 flyers left. What is the exptected time it would take to run out of flyers?
a) What is the probability that it takes more than 10 minutes for the volunteer to give out the first flyer? b) What is the expected number of flyers the volunteer would distribute in 6 hours? c) Suppose the volunteer has 3 flyers left. What is the exptected time it would take to run out of flyers?
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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![Axloms of Probablity
Also Note
1. P(8)-1
2. For any event E, 0S P(E)s1
For any two events A and B,
P(A) - P(AN B) + P(ANB)
3. For any two mutually exclusive events,
and
P(EUF) - P(E) + P(F)
P(AN B) - P(A|B)P(B).
Addition Rule
Events A and B are Independent if:
P(EUF) = P(E) + P(F) - P(En F)
P(A|B) = P(A)
Conditional Probablity
or
P(B|A) -
P(ANB) - P(A)P(B).
PLAN)
Bayes' Theorem:
Total Probablity Rule
P(A|B)P(B)
P(B|A) = PLALBPB) + P(AB)P(B")
P(A) - P(A|B)P(B) + P(A|B')P(B')
Similarly,
Similarly,
P(A) -P(A|E,)P(E)) + P(A|E)P(E)+
...+ P(A|E)P(E)
P(B|E)P(E)
P(E|B) - PIBIE PE) + P(BE PE)+...+ P(B\E)P(E.)
Probability Mass and Density Functions
If X is a discrete r.v:
Cumulative Distribution Function
• F(z) = P(X sz)
P(X = 2) = f(z)
• lim,- F() -0
Es(2) =1 (total probability)
• lim,e F(z) = 1
If X is a continuous r.v.:
P(X = z) = 0
• F(z) = " /(v)dy if X is a contimuous r.v.
S(2)dz =1 (total probability)
• F(z) = E,sz f(z) if X is a discrete r.v.
• P(a < X Sb) - F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• E[X) = E, z/(z) if X is a discrete r.v.
• E[h(X)] =E. h(x)f(x) if X is a discrete r.v.
• Eh(X)) = h(z)/(z)dz if X is a continu-
• E[X] = z/(z)dr if X is a continuous r.v.
ous r.v.
• Var(X) = E[Xx] – E[X]?
• E(aX + 6) = aEX] + 6
• Var(aX + b) = a?Var(X)
%3D
• Var(X) = E[(X - E[X])?]
Derivatives and Integrals of Common Functions
• = aea
de
• Sea" dz =
• Sre*dr = e"I- fe*dz = ze" - e (using integration by parts)
dinz
• S !dz = In(z)
Common Discrete Distributions
• X - Bernoulli(p),
if z = 1;
f(z) =
|1-p ifz 0' EX] = p, Var(X) = p(1 – p).
• X- Geometric(p),
f(2) = (1– p)--'p, z E {1,2,..}, E[X] = }, Var(X) = .
Geometric Series: Eg = , for 0 < q < 1
• X - Binomial(n, p),
f(z) = (E) (1– p)"-p*, I € {0, 1,.., n},
E[X] = np, Var(X) = mp(1 – p).
%3D
• X- Negative Binomial(r, p),
f(z) = ()(1 – p)*-"p", E[X] = ;, 1 € {r,r+1,..}, Var(X) = p),
%3D
• X - Hypergeometric(n, M, N),
f(z) =
,
E[X] = n, Var(X) = N=n(1-).
%3D
• X ~ Poisson(At),
f(z) = A0", z e {0, 1, .}, E[X] = At, Var(X) = At.
Common Continuous Distributions
• X - Exponential(A),
f(z) = de-A, z E [0, 00) E[X] = }, Var(X)= .
• X- Erlang(r, A),
f(z) = A' , zE (0, 00), E[X] = 5, Var(X) = .
Suppose that Duke Energy mu](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F404f3a78-f730-4773-ad7b-fe6612b342a7%2Fcde05519-af9a-433d-8b85-d4c9f81bc649%2Fjb9wyv5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Axloms of Probablity
Also Note
1. P(8)-1
2. For any event E, 0S P(E)s1
For any two events A and B,
P(A) - P(AN B) + P(ANB)
3. For any two mutually exclusive events,
and
P(EUF) - P(E) + P(F)
P(AN B) - P(A|B)P(B).
Addition Rule
Events A and B are Independent if:
P(EUF) = P(E) + P(F) - P(En F)
P(A|B) = P(A)
Conditional Probablity
or
P(B|A) -
P(ANB) - P(A)P(B).
PLAN)
Bayes' Theorem:
Total Probablity Rule
P(A|B)P(B)
P(B|A) = PLALBPB) + P(AB)P(B")
P(A) - P(A|B)P(B) + P(A|B')P(B')
Similarly,
Similarly,
P(A) -P(A|E,)P(E)) + P(A|E)P(E)+
...+ P(A|E)P(E)
P(B|E)P(E)
P(E|B) - PIBIE PE) + P(BE PE)+...+ P(B\E)P(E.)
Probability Mass and Density Functions
If X is a discrete r.v:
Cumulative Distribution Function
• F(z) = P(X sz)
P(X = 2) = f(z)
• lim,- F() -0
Es(2) =1 (total probability)
• lim,e F(z) = 1
If X is a continuous r.v.:
P(X = z) = 0
• F(z) = " /(v)dy if X is a contimuous r.v.
S(2)dz =1 (total probability)
• F(z) = E,sz f(z) if X is a discrete r.v.
• P(a < X Sb) - F(b) – F(a)
Expected Value and Variance
Expected Value of a Function of a RV
• E[X) = E, z/(z) if X is a discrete r.v.
• E[h(X)] =E. h(x)f(x) if X is a discrete r.v.
• Eh(X)) = h(z)/(z)dz if X is a continu-
• E[X] = z/(z)dr if X is a continuous r.v.
ous r.v.
• Var(X) = E[Xx] – E[X]?
• E(aX + 6) = aEX] + 6
• Var(aX + b) = a?Var(X)
%3D
• Var(X) = E[(X - E[X])?]
Derivatives and Integrals of Common Functions
• = aea
de
• Sea" dz =
• Sre*dr = e"I- fe*dz = ze" - e (using integration by parts)
dinz
• S !dz = In(z)
Common Discrete Distributions
• X - Bernoulli(p),
if z = 1;
f(z) =
|1-p ifz 0' EX] = p, Var(X) = p(1 – p).
• X- Geometric(p),
f(2) = (1– p)--'p, z E {1,2,..}, E[X] = }, Var(X) = .
Geometric Series: Eg = , for 0 < q < 1
• X - Binomial(n, p),
f(z) = (E) (1– p)"-p*, I € {0, 1,.., n},
E[X] = np, Var(X) = mp(1 – p).
%3D
• X- Negative Binomial(r, p),
f(z) = ()(1 – p)*-"p", E[X] = ;, 1 € {r,r+1,..}, Var(X) = p),
%3D
• X - Hypergeometric(n, M, N),
f(z) =
,
E[X] = n, Var(X) = N=n(1-).
%3D
• X ~ Poisson(At),
f(z) = A0", z e {0, 1, .}, E[X] = At, Var(X) = At.
Common Continuous Distributions
• X - Exponential(A),
f(z) = de-A, z E [0, 00) E[X] = }, Var(X)= .
• X- Erlang(r, A),
f(z) = A' , zE (0, 00), E[X] = 5, Var(X) = .
Suppose that Duke Energy mu
Transcribed Image Text:3. A volunteer is distributing flyers at a street corner for a comedy show. The number of people who
accept a flyer as they pass by follows a Poisson distribution with the rate of u = 4 people per hour.
a) What is the probability that it takes more than 10 minutes for the volunteer to give out the first
flyer?
b) What is the expected number of flyers the volunteer would distribute in 6 hours?
c) Suppose the volunteer has 3 flyers left. What is the exptected time it would take to run out of
flyers?
d) What is the probability that it would take more than an hour to distribute the last 3 flyers?
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