Flashcards
Chapter 4
Cards in this deck (26)
- 1What is a random variable (rv)?A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables, discrete and continuous.
- 2What is P(X=x) for any possible value x, for a continour rv X?0
It must be an interval if X is continous - 3What is a Bernouli random variableAny random variable whose possible values are 0 and 1
- 4What is probability mass function (pmf) of a discrete rv X?f(x) gives the probability of each possible value x of X
f(x) = P(X=x) for ALL possible values of X. - 5What are the properties of pmf of a discrete function?1) f(x) >= 0 for all x in X
2) Σ f(x) = 1 - 6What is cumulative distribution function (cdf)?F(x) = P(X ≤ x) = Σ f(k) for all real x, not just discrete
- 7What is the mean/expectation/expected value of a discrete rv X?long run average of X over hypothetical repetitions of experiment
μ = E[X] = Σ x*f(x) - 8What is the variance value of a discrete rv X?σ² = Var(x) = E[(x-μ)²] = Σ (x-μ)² * f(x)
or
σ² = Var(x) = E[X²] - E[X]² - 9What is probability density function (pmf) of a continuous rv X?
- 10What are the properties of pdf of a continuous function?1) f(x) >= 0 for all x
2) ∫ f(x)dx = 1
from -∞ to ∞ - 11Explain in words what cdf of a continuous rv X is.Probability that X takes on a value in the interval [a,b] is the area above this interval and under the graph of the density function as given in the figure
- 12What is the cumulative distribution function (cdf) of a continuous rv XF(x) = P(X≤x) = ∫ f(t) dt (from -∞ to x) for all x
- 13Write P(X>a) in two ways1) 1 - F(a)
2) 1 - P(X≤a)
also, 1 - Fₓ(a) - 14Write P(a1) F(b) - F(a)
2) P(X≤b) - P(X≤a) - 15f(x) → F(x) by ________ F(x) → f(x) by ________integration
differentiation - 16How do you find median, Q1, Q3, and IQR from f(x)?1) Find F(x)
2) F(x) = 0.5 or 0.25 or 0.75 and solve for x
3) IQR = Q3 - Q1 - 17What is the mean/expectation/expected value of a continuous rv X?μ = E[X] = ∫ x*f(x)
(from -∞ to ∞) - 18What is the variance value of a continuous rv X?σ² = Var(x) = E[(x-μ)²] = ∫ (x-μ)² * f(x)
(from -∞ to ∞)
or
σ² = Var(x) = E[X²] - E[X]² - 19Uniform Distribution: Notation, interval, pdf, mean, variance?Notation: X~U(a,b)
Interval/support: [a,b]
Pdf: f(x) = 1/(b-a) ; a≤x≤b
Mean: (a+b)/2
Variance: (b-a)²/12
cdf: (x-a)/(b-a) for x = [a,b)
1 for x≥b - 20Exponential Distribution: interval, pdf, mean, variance?Interval/support: [0,∞]
Pdf: f(x) = λe^(-λx) ; x≥0, λ>0
Mean: 1/λ
Variance: 1/λ²
cdf: 1 − e^(−λx) - 21Properties of Mean: E[aX + b] = ______ for all a and b E[X + Y] = ______ for all X and Y E[XY] = ______ for independent X, YaE[X] + b
E[X] + E[Y]
E[X] * E[Y]
(E of constant is constant, E distributes) - 22Properties of Variance: Var(Ax+bY) = _____ V(b) = _____ Cov(X,Y) = _____ when X,Y independenta²Var(X) + b²Var(Y) + 2abCov(X,Y)
0
0
(Var(X+Y) and Var(X-Y) is same when cov=0) - 23Explain sum of independent random variables.Y = a1X1 + a2X2 + ... + anXn
Then,
E[Y] = a1E[X1] + a2E[X2] + ... + anE[Xn]
Var[Y] = a1²Var(X1) + a2²Var(X2) + ... + an²Var(Xn)
If all X's have common μ and σ², then
E[Y] = (a1 +a2 + ... + an)μ
Var(Y) = (a1 +a2 + ... + an)σ² - 24Explain average of independent random variables.X̄ = (X1 + X2 + ... + Xn)/n
E[X̄] = 1/n{E[X1] + E[X2] + ... + E[Xn]}
Var(X̄) = 1/n²{Var(X1) + Var(X2) + ... + var(Xn)}
If all X's have common μ and σ², then,
E[X̄] = μ
Var(X̄) = σ²/n - 25When is the maximum of independent rv's used?- lifetime of a system connected in parallel
- maximum flood level of a river
V = max{X1, X2, ... Xn}
each X has pdf fₓ(x) and cdf Fₓ(x)
V has cdf Fᵥ(v) = P(V≤v) = P(X1 - 26When is the minimum of independent rv's used?- lifetime of a system connected in series
- minimum flood level of a river
U = min{X1, X2, ... Xn}
each X has pdf fₓ(x) and cdf Fₓ(x)
V has cdf Fᵤ(u) = P(U≤u) = 1-P(U>u) = 1- P(X1>u)... = 1 - [1-Fₓ(u)]ⁿ
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- Flashcards on Chapter 4, created by Manisha Walia on 29/05/2018.
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