Exam 2 (pt. 2) Flashcards
Chapter 5 Content
An r.v. has a continuous distribution if its CDF is ___________.
We also allow there to be endpoints (or finitely many points) where the CDF is continuous but not differentiable, as long as . . .
differentiable
the CDF is differentiable everywhere else
Let X be a continuous r.v. with PDF f . Then the CDF of X is given by . . .
Probabilities for continuous random variables are specified by__________. This leads us to the special rule that P(X = x) = _______ for continuous random variables
the area under a curve
0
For sets of the form [a, b], (a, b], [a, b), (a, b), the CDF is dereived from the PMF by . . .
To get a desired probability for a continuous r.v. you . . .
integrate the PDF over the desired range
The PDF f of a continuous r.v. must satisfy the following two criteria:
The expected value (also called the expectation or mean) of a continuous r.v. X with PDF f is . . .
If X is a continuous r.v. with PDF f and g is a function from R to R, then E(g(X))= ___
To go from PDF to CDF of a continuous distribution you _______
Integrate f(x)
What is the power rule for differentiating a function?
To go from a CDF to a PDF of a continuous distribution you _______
differentiate F(x)
What is the power rule for integrating a function?
To find the probability of P(a < x < b) you ____________ the _________ on (a,b)
integrate, PDF
A continuous r.v. U is said to have the Uniform distribution on the interval (a, b) if its PDF is . . .
The CDF of the continuous uniform is . . .
The uniform distribution is denoted as ________. The standard uniform is _______
U ~ Unif(a , b)
U ~ Unif(0 , 1)
What is E(U) of U ~ Unif(a , b)?
E(U) = (a + b)/2
What is Var(U) of U ~ Unif(a , b)?
(b - a)^2/12
A continuous r.v. X is said to have the Exponential distribution with parameter λ > 0 if its PDF is . . .
A continuous r.v. X is said to have the Exponential distribution with parameter λ > 0 if its CDF is . . .
The exponential distribution is denoted as . . .
X ~ Exp(λ)
If X ~ Exp(1), then Y ~ λ^-1X ~ ________.
What would the converse be?
What is E(X) and Var(X) given that X ~ Exp(λ)?
A continuous distribution is said to have the memoryless property if a random variable X from that distribution satisfies . . .
The memoryless property is a very special property of the Exponential distribution because . . .
no other continuous distribution on
(0, ∞) is memoryless
A continuous r.v. Z is said to have the standard Normal distribution if its PDF ϕ is given by . . .
How is the standard, normal distribution notated?
Z ~ N(0,1)
What is the CDF of the standard, normal distribution?
the central limit theorem says that under very weak assumptions, the sum of a large number of i.i.d. random
variables has an . . .
approximately Normal distribution, regardless of the distribution of the individual r.v.s.
What are the three important symmetry properties that can be deduced from the standard Normal PDF and CDF?
For a standard, normal distribution, the E(Z) = _______
For a standard, normal distribution, the Var(Z) = ________
If Z ∼ N(0, 1) then X = μ + σZ has . . .
a normal distribution with mean μ and variance σ^2, for any μ ∈ R, and σ^2 with σ > 0.
By linearity, E(X) of a normal distribution = _________
By linearity, Var(X) of a normal distribution = _________
For X ∼ N(μ, σ2) the standardized version of X is given by . . .
For X ∼ N(μ, σ2) the standardized CDF of X is . . .
For X ∼ N(μ, σ2) the standardized PDF of X is . . .
Let X ∼ N(−1, 4). What is P(|X| < 3), exactly in terms of Φ?
If X ∼ N(μ, σ2) then . . .
1. P(|X − μ| < σ) ≈
2. P(|X − μ| < 2σ) ≈
3. P(|X − μ| < 3σ) ≈
This is commonly known as _____ rule
- P(|X − μ| < σ) ≈ 0.68
- P(|X − μ| < 2σ) ≈ 0.95
- P(|X − μ| < 3σ) ≈ 0.997
68-95-99.7% rule
Let X ∼ N(−1, 4). What is P(|X| < 3), approximately?
Given that X ~ Exp(λ) what is a rate parameter versus a scale parameter?
Rate parameter = λ
Scale Parameter = 1/λ
What is P(c < x < d) for X~Unif(a,b)?
(d-c/b-a)
How do you find percentile of a normal distribution?
X = μ + (z x σ)
where z corresponds to the Z score (X − μ /σ) of the desired percentile
