Lecture 1-2
... (3) P A = 1 − P Ac for A ∈ S Proof : Since A and Ac are disjoint and their union is Ω we have by (2) P A + P Ac = P (Ω) = 1 (Remember : Ac ∈ S since A ∈ S). (4) If E ⊆ F for two sets E, F ∈ S, then P E ≤ P F (monotonicity) Proof : Since E ⊆ F we have two disjoint sets E and F \E whose union is F . H ...
... (3) P A = 1 − P Ac for A ∈ S Proof : Since A and Ac are disjoint and their union is Ω we have by (2) P A + P Ac = P (Ω) = 1 (Remember : Ac ∈ S since A ∈ S). (4) If E ⊆ F for two sets E, F ∈ S, then P E ≤ P F (monotonicity) Proof : Since E ⊆ F we have two disjoint sets E and F \E whose union is F . H ...
Powerpoints
... • Rest of chains are “heated”: move faster across valleys • Each turn the cold and warm chains may swap position (swap probability is proportional to ratio between heights) More peaks will be visited More chains means better chance of visiting all important peaks, but each additional ...
... • Rest of chains are “heated”: move faster across valleys • Each turn the cold and warm chains may swap position (swap probability is proportional to ratio between heights) More peaks will be visited More chains means better chance of visiting all important peaks, but each additional ...
Ch. 4-6 PowerPoint Review
... Our goal in statistics is often to answer a question about a population using information from a sample. to do this, the sample must be representative of the population in question ...
... Our goal in statistics is often to answer a question about a population using information from a sample. to do this, the sample must be representative of the population in question ...
Probability Application Set: Name: A poll found that 46% of
... 13. An urn contains 3 red chips, 2 blue chips, and 5 white chips. A chip is selected and its color noted. Then it is replaced. A second chip is selected and it color noted. Find the probability of each: a. Selecting 2 blue chips ...
... 13. An urn contains 3 red chips, 2 blue chips, and 5 white chips. A chip is selected and its color noted. Then it is replaced. A second chip is selected and it color noted. Find the probability of each: a. Selecting 2 blue chips ...
A∪ A∩
... e. Define D to be a subset of A. 2. A teacher randomly chooses a twoperson team from a group of four students. The first person chosen will be the presenter and the second person will be the researcher. Two of the students, Amir and Aaron, are boys. The other two students, Caitlin and Deni ...
... e. Define D to be a subset of A. 2. A teacher randomly chooses a twoperson team from a group of four students. The first person chosen will be the presenter and the second person will be the researcher. Two of the students, Amir and Aaron, are boys. The other two students, Caitlin and Deni ...
I Agree
... makes sense only if the number of possible outcomes is finite. We will only consider these cases. ...
... makes sense only if the number of possible outcomes is finite. We will only consider these cases. ...
