Exam (elaborations)
Che 505 Final Exam Part 4 (Oral) With Complete Solutions Latest Update
- Course
- CHE 505
- Institution
- Arizona State University
Che 505 Final Exam Part 4 (Oral) With Complete Solutions Latest Update
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Solution 2024/2025Pepper
Che 505 Final Exam Part 4 (Oral) With
Complete Solutions Latest Update
Random Process ANS✔✔ process where what happens is characterized by
uncertainty
Outcomes ANS✔✔ potential result of performing an experiment
Ω- all possible outcomes of a random process
event ANS✔✔ any non empty subset of outcomes that can occur
ie rolling an off number when rolling a die
Probability metric properties ANS✔✔ for any event A we can define a
probability associated with event that tells us likelihood that event occurs -
pr(A)
probability matrix properties
Pr(ø)=0
in an empty set there is no change that no outcome occurs
pr(A)≥0 no event has negative probability
, Solution 2024/2025
Pepper
if A1 and A2 are disjoint events (share no common outcomes)
Pr(A1 U A2) = Pr(A1) + Pr(A2)
Pr(Ω) = 1 law of total probability
complimentary event ANS✔✔ event A, its complement A^c is set of all
outcomes where A does not occur
A and A^c are disjoint events that satisfy A U A^c = Ω
Pr(A U A^c) = Pr (Ω)
Pr(A) + Pr(A^c) = 1
Pr(A^c) = 1-Pr(A)
random variable ANS✔✔ set of all uncertian outcomes for a random process
and their probabilities
discrete random variable ANS✔✔ set of all possible outcomes is finite por
countably infinite in size
rolling a dice
continuous random variable ANS✔✔ set of all possible outcomes consist of a
continumn of oncountably infinite possible results
spin a spinner and measure and angle
, Solution 2024/2025
Pepper
Probability mass funciton ANS✔✔ discrete random variable is characterized
by a probably mass function
∑px(X) = 1
consequence of law of total probabilty
any pmf must satisfy this property
Probability density function ANS✔✔ continuous random variable is
characterized by a probability density function
use to determine likelihood x is between a and b
Pr(a≤x≤b) = ∫px(X)dx
∫px(X)dx = 1
integral from -∞ to ∞
consequence of law of total probability
any PDF must satisfy this conditon
common discrete random variable distributions ANS✔✔ uniform, bernoili
(binomial), emperical
, Solution 2024/2025
Pepper
common continuous random variable distributions ANS✔✔ uniform,
gaussian (normal)
Expectation and its properties ANS✔✔ expectation of a random variable or
function is defined as an average of a function over all possible outcomes Ω
weighted by corresponding PMF/PDF
E[aX + bY] = aE[X] + bE[Y]
E[a] = a
E[E[X]] = E[X]
expectation related to law of large numbers: If I sample a random process
many times, average of observed f(xk) = E[f(xk)]
expectation of discrete RV ANS✔✔ E[f(x)] = ∑px(x)f(x)
expectation of a continuous random variable ANS✔✔ E[f(x)] = ∫px(X)f(x)dx
momements ANS✔✔ µp = E[x^p]
knowing the momenents of a random variable gives a lot of information
about underlying distribution
most information is imbedded in the lowest order moments
first moment ANS✔✔ µ1 = E[x]
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