test2qna Flashcards
What is a random variable?
A random variable is a function that assigns a real number to each outcome in a sample space.
What are the two types of random variables?
Discrete random variables take countable (often finite) values, while continuous random variables take values in an interval (or collection of intervals).
What is the probability mass function (PMF)?
For a discrete random variable X, the PMF p(x) gives P(X = x) for each value x.
What is the probability density function (PDF)?
For a continuous random variable X, the PDF f(x) satisfies P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx, with f(x) ≥ 0 and ∫₋∞∞ f(x) dx = 1.
What is the cumulative distribution function (CDF)?
The CDF F(x) = P(X ≤ x) gives the probability that a random variable X takes on a value less than or equal to x.
What are the properties of the CDF?
The CDF is non-decreasing, right-continuous, and satisfies limₓ→₋∞ F(x) = 0 and limₓ→∞ F(x) = 1.
How is the expected value (mean) defined?
For a discrete random variable, E[X] = Σ x · P(X=x); for a continuous variable, E[X] = ∫₋∞∞ x · f(x) dx.
How is the variance defined?
Variance is Var(X) = E[(X – E[X])²] = E[X²] – (E[X])².
What is a moment generating function (MGF)?
The MGF of X is Mₓ(t) = E[e^(tX)], which, by differentiating with respect to t and evaluating at t = 0, can be used to derive moments (mean, variance, etc.).
What is the Law of the Unconscious Statistician (LOTUS)?
LOTUS states that E[g(X)] = Σ g(x)P(X=x) for discrete variables or E[g(X)] = ∫ g(x) f(x) dx for continuous variables—no need to find the distribution of g(X) first.
What is a characteristic function?
The characteristic function φ_X(t) = E[e^(itX)] uniquely determines the distribution of X and is useful for studying convergence in distribution.
What is linearity of expectation?
For any random variables X and Y and constants a, b, E[aX + bY] = aE[X] + bE[Y] (no independence required).
What is a Bernoulli random variable?
A Bernoulli random variable X takes two values: 0 (failure) with probability 1-p and 1 (success) with probability p, where 0 ≤ p ≤ 1.
What is the PMF of a Bernoulli random variable?
P(X=0) = 1 - p and P(X=1) = p.
What are the expectation and variance of a Bernoulli random variable?
E[X] = p and Var(X) = p(1 - p).
What is a Binomial random variable?
A Binomial random variable X counts the number of successes in n independent Bernoulli trials with success probability p.
What is the PMF of a Binomial random variable?
P(X=i) = C(n, i) · p^i · (1-p)^(n-i), for i = 0, 1, …, n.
What are the expectation and variance of a Binomial(n, p) distribution?
E[X] = np and Var(X) = np(1 - p).
What is a Poisson random variable?
A Poisson random variable X takes values 0, 1, 2, … with PMF: P(X=i) = (e^(–λ) · λ^i) / i! where λ > 0 is the rate parameter.
What are the expectation and variance of a Poisson(λ) distribution?
E[X] = λ and Var(X) = λ.
What is an Exponential random variable?
An Exponential random variable with parameter λ models waiting times and has PDF: f(x) = λe^(–λx) for x ≥ 0.
What are the expectation and variance of an Exponential(λ) distribution?
E[X] = 1/λ and Var(X) = 1/λ².
What is the memoryless property of the Exponential distribution?
P(X > s + t
What is the Normal distribution?
A normal (or Gaussian) random variable X ~ N(µ, σ²) has PDF: f(x) = (1/√(2πσ²))e^(–(x–µ)²/(2σ²)), defined for all x ∈ ℝ.
What are the expectation and variance of a Normal(µ, σ²) distribution?
E[X] = µ and Var(X) = σ².
How do you standardize a normal variable?
For X ~ N(µ, σ²), Z = (X – µ)/σ transforms X to the standard normal distribution, Z ~ N(0, 1).
What is the Gamma distribution?
A Gamma random variable with parameters α (shape) and θ (scale) has PDF: f(x) = (1/(Γ(α)θ^α)) x^(α–1)e^(–x/θ) for x > 0.
What are the expectation and variance of a Gamma(α, θ) distribution?
E[X] = αθ and Var(X) = αθ².
What is the relationship between the Exponential and Gamma distributions?
The exponential distribution is a special case of the Gamma distribution with α = 1 and θ = 1/λ (when the exponential parameter is λ).
What is the sum of independent Poisson random variables?
If X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are independent, then X + Y ~ Poisson(λ₁ + λ₂).
What is the Central Limit Theorem (CLT)?
The CLT states that the sum (or average) of a large number of independent, identically distributed random variables (with finite mean and variance) approximates a normal distribution, regardless of the original distribution.
What is the transformation rule for a continuous random variable?
If Y = g(X) is a one-to-one transformation with inverse g⁻¹, then the PDF of Y is given by f_Y(y) = f_X(g⁻¹(y)) ·
Why is the standard normal distribution important in statistics?
The standard normal distribution is the basis for many inferential techniques, such as hypothesis testing and confidence interval estimation, and it arises naturally via the CLT.
What defines a continuous random variable, and what two properties must its probability density function (PDF) satisfy?
A continuous random variable XX is defined by a nonnegative PDF f(x)f(x), where P(X∈B)=∫Bf(x)dxP(X∈B)=∫Bf(x)dx for any subset B⊆RB⊆R. The PDF must satisfy: (1) f(x)≥0f(x)≥0 for all xx, (2) ∫−∞∞f(x)dx=1∫−∞∞f(x)dx=1.
How do you find the constant CC for the PDF f(x)=C(6x−x2)f(x)=C(6x−x2) on (0,6)(0,6)?
Solve ∫06C(6x−x2)dx=1∫06C(6x−x2)dx=1. Integration yields C[3x2−x33]06=36C=1C[3x2−3x3]06=36C=1, so C=136C=361.
For f(x)=136(6x−x2)f(x)=361(6x−x2), what is the probability X>3X>3?
Compute P(X>3)=∫36136(6x−x2)dxP(X>3)=∫36361(6x−x2)dx. Result: 136[3x2−x33]36=12361[3x2−3x3]36=21.
Battery lifetime follows f(x)=100x2f(x)=x2100 for x>100x>100. What’s the probability a battery fails within 150 hours, and how is this used for 2 out of 5 batteries?
Probability of failure within 150h: ∫100150100x2dx=13∫100150x2100dx=31. For 2/5 failures: Use the binomial formula: (52)(13)2(23)3≈0.329(25)(31)2(32)3≈0.329.
How does scaling a continuous RV XX to Y=5XY=5X affect its PDF?
If XX has PDF fX(x)fX(x), then Y=5XY=5X has PDF fY(y)=15fX(y5)fY(y)=51fX(5y). This follows from transforming the CDF: FY(y)=FX(y5)FY(y)=FX(5y).
Calculate E[X]E[X] for f(x)=3x2f(x)=3x2 on [0,1][0,1].
E[X]=∫01x⋅3x2dx=3∫01x3dx=3⋅14=34E[X]=∫01x⋅3x2dx=3∫01x3dx=3⋅41=43.
Derive the variance for f(x)=3x2f(x)=3x2 on [0,1][0,1].
First compute E[X2]=∫01x2⋅3x2dx=35E[X2]=∫01x2⋅3x2dx=53. Then Var(X)=E[X2]−(E[X])2=35−(34)2=380Var(X)=E[X2]−(E[X])2=53−(43)2=803.
If Var(X)=3.5Var(X)=3.5, what is Var(2X−7)Var(2X−7)?
Variance scales with a2a2: Var(2X−7)=22⋅Var(X)=4×3.5=14Var(2X−7)=22⋅Var(X)=4×3.5=14. The constant −7−7 does not affect variance.
Uniform RV X∼(3,8)X∼(3,8): What is P(X<5)P(X<5)?
For uniform (α,β)(α,β), P(a<X<b)=b−aβ−αP(a<X<b)=β−αb−a. Here, P(3<X<5)=5−38−3=25P(3<X<5)=8−35−3=52.
What are the expectation and variance formulas for a uniform RV (α,β)(α,β)?
Expectation: E[X]=α+β2E[X]=2α+β. Variance: Var(X)=(β−α)212Var(X)=12(β−α)2. Example: For (3,8)(3,8), E[X]=5.5E[X]=5.5, Var(X)=2512Var(X)=1225.
Passengers arrive uniformly between 7:00-7:30. What’s the probability of waiting less than 5 minutes for a bus?
Buses arrive every 15 minutes. Waiting <5 minutes occurs if arrival is in [10,15)[10,15) or [25,30)[25,30). Probability: 530+530=13305+305=31.
Why does E[X1+⋯+Xn]=E[X1]+⋯+E[Xn]E[X1+⋯+Xn]=E[X1]+⋯+E[Xn], even for dependent variables?
Linearity of expectation holds regardless of dependence. Example: For dice rolls, E[sum]=10×3.5=35E[sum]=10×3.5=35. Dependence affects variance, not expectation.
How is the expected number of white balls selected in a hypergeometric experiment derived?
Use indicator variables: Let Xi=1Xi=1 if the ii-th white ball is chosen. Then E[X]=∑i=1mE[Xi]=m⋅nNE[X]=∑i=1mE[Xi]=m⋅Nn, since P(Xi=1)=nNP(Xi=1)=Nn.
In the hat-matching problem, why is the expected number of matches 1?
Let Ii=1Ii=1 if person ii gets their hat. E[Ii]=1NE[Ii]=N1, so E[X]=∑i=1N1N=1E[X]=∑i=1NN1=1. Linearity simplifies complex dependencies.
What does covariance measure, and how is it calculated?
Covariance measures joint variability: Cov(X,Y)=E[(X−E[X])(Y−E[Y])]=E[XY]−E[X]E[Y]Cov(X,Y)=E[(X−E[X])(Y−E[Y])]=E[XY]−E[X]E[Y]. If independent, Cov(X,Y)=0Cov(X,Y)=0.
How is the variance of a sum of RVs affected by covariance?
Var(X1+⋯+Xn)=∑Var(Xi)+2∑i<jCov(Xi,Xj)Var(X1+⋯+Xn)=∑Var(Xi)+2∑i<jCov(Xi,Xj). For independence, covariance terms vanish, leaving ∑Var(Xi)∑Var(Xi).
If Y=a+bXY=a+bX, why is ρ(X,Y)=1ρ(X,Y)=1 for b>0b>0?
Correlation ρ(X,Y)=Cov(X,Y)Var(X)Var(Y)ρ(X,Y)=Var(X)Var(Y)Cov(X,Y). Here, Cov(X,Y)=bVar(X)Cov(X,Y)=bVar(X) and Var(Y)=b2Var(X)Var(Y)=b2Var(X), so ρ=b∥b∥=1ρ=∥b∥b=1.
For 10 fair dice rolls, how is the total variance calculated?
Single die variance: Var(Xi)=3512Var(Xi)=1235. Total variance: 10×3512=1756≈29.1710×1235=6175≈29.17. Independence ensures variances add directly.
