4. Suppose that X has an exponential distribution with a mean of 8. Determine the following: (a) P(X < 4) (b) P(X < 16|X > 12) (c) Compare the results in parts (a) and (b) and comment on the role of memoryless property.
4. Suppose that X has an exponential distribution with a mean of 8. Determine the following: (a) P(X < 4) (b) P(X < 16|X > 12) (c) Compare the results in parts (a) and (b) and comment on the role of memoryless property.
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Transcribed Image Text:4. Suppose that X has an exponential distribution with a mean of 8. Determine the following:
(a) P(X < 4)
(b) P(X < 16|X > 12)
(c) Compare the results in parts (a) and (b) and comment on the role of memoryless property.
![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%2Fc97efd08-ba00-4a2f-9848-0e0f076bfc1e%2F1rn5yaw_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
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