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a) Compute the value of C. b) Compute the cumulative distribution function of X c) Compute P(X > 0.2). d) Computo Y1

a) Compute the value of C. b) Compute the cumulative distribution function of X c) Compute P(X > 0.2). d) Computo Y1

A First Course in Probability (10th Edition)
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 show all your work! Attached is the formula sheet
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: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
PLEASE SHOW ALL YOUR WORK 1. Suppose X is a random variable with the density function f(x) = C(4x – x²) for 0 <r< 1. a) Compute the value of C. b) Compute the cumulative distribution function of X c) Compute P(X > 0.2). d) Compute E[X]. 1..4:
Transcribed Image Text:PLEASE SHOW ALL YOUR WORK 1. Suppose X is a random variable with the density function f(x) = C(4x – x²) for 0 <r< 1. a) Compute the value of C. b) Compute the cumulative distribution function of X c) Compute P(X > 0.2). d) Compute E[X]. 1..4:
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