test2qna

Question 1

What is a random variable?

Answer 1

A random variable is a function that assigns a real number to each outcome in a sample space.

Question 2

What are the two types of random variables?

Answer 2

Discrete random variables take countable (often finite) values, while continuous random variables take values in an interval (or collection of intervals).

Question 3

What is the probability mass function (PMF)?

Answer 3

For a discrete random variable X, the PMF p(x) gives P(X = x) for each value x.

Question 4

What is the probability density function (PDF)?

Answer 4

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.

Question 5

What is the cumulative distribution function (CDF)?

Answer 5

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.

Question 6

What are the properties of the CDF?

Answer 6

The CDF is non-decreasing, right-continuous, and satisfies limₓ→₋∞ F(x) = 0 and limₓ→∞ F(x) = 1.

Question 7

How is the expected value (mean) defined?

Answer 7

For a discrete random variable, E[X] = Σ x · P(X=x); for a continuous variable, E[X] = ∫₋∞∞ x · f(x) dx.

Question 8

How is the variance defined?

Answer 8

Variance is Var(X) = E[(X – E[X])²] = E[X²] – (E[X])².

Question 9

What is a moment generating function (MGF)?

Answer 9

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.).

Question 10

What is the Law of the Unconscious Statistician (LOTUS)?

Answer 10

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.

Question 11

What is a characteristic function?

Answer 11

The characteristic function φ_X(t) = E[e^(itX)] uniquely determines the distribution of X and is useful for studying convergence in distribution.

Question 12

What is linearity of expectation?

Answer 12

For any random variables X and Y and constants a, b, E[aX + bY] = aE[X] + bE[Y] (no independence required).

Question 13

What is a Bernoulli random variable?

Answer 13

A Bernoulli random variable X takes two values: 0 (failure) with probability 1-p and 1 (success) with probability p, where 0 ≤ p ≤ 1.

Question 14

What is the PMF of a Bernoulli random variable?

Answer 14

P(X=0) = 1 - p and P(X=1) = p.

Question 15

What are the expectation and variance of a Bernoulli random variable?

Answer 15

E[X] = p and Var(X) = p(1 - p).

Question 16

What is a Binomial random variable?

Answer 16

A Binomial random variable X counts the number of successes in n independent Bernoulli trials with success probability p.

Question 17

What is the PMF of a Binomial random variable?

Answer 17

P(X=i) = C(n, i) · p^i · (1-p)^(n-i), for i = 0, 1, …, n.

Question 18

What are the expectation and variance of a Binomial(n, p) distribution?

Answer 18

E[X] = np and Var(X) = np(1 - p).

Question 19

What is a Poisson random variable?

Answer 19

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.

Question 20

What are the expectation and variance of a Poisson(λ) distribution?

Answer 20

E[X] = λ and Var(X) = λ.

Question 21

What is an Exponential random variable?

Answer 21

An Exponential random variable with parameter λ models waiting times and has PDF: f(x) = λe^(–λx) for x ≥ 0.

Question 22

What are the expectation and variance of an Exponential(λ) distribution?

Answer 22

E[X] = 1/λ and Var(X) = 1/λ².

Question 23

What is the memoryless property of the Exponential distribution?

Answer 23

P(X > s + t

Question 24

What is the Normal distribution?

Answer 24

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.

Question 25

What are the expectation and variance of a Normal(µ, σ²) distribution?

Answer 25

E[X] = µ and Var(X) = σ².

Question 26

How do you standardize a normal variable?

Answer 26

For X ~ N(µ, σ²), Z = (X – µ)/σ transforms X to the standard normal distribution, Z ~ N(0, 1).

Question 27

What is the Gamma distribution?

Answer 27

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.

Question 28

What are the expectation and variance of a Gamma(α, θ) distribution?

Answer 28

E[X] = αθ and Var(X) = αθ².

Question 29

What is the relationship between the Exponential and Gamma distributions?

Answer 29

The exponential distribution is a special case of the Gamma distribution with α = 1 and θ = 1/λ (when the exponential parameter is λ).

Question 30

What is the sum of independent Poisson random variables?

Answer 30

If X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are independent, then X + Y ~ Poisson(λ₁ + λ₂).

Question 31

What is the Central Limit Theorem (CLT)?

Answer 31

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.

Question 32

What is the transformation rule for a continuous random variable?

Answer 32

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)) ·

Question 33

Why is the standard normal distribution important in statistics?

Answer 33

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.

Question 34

What defines a continuous random variable, and what two properties must its probability density function (PDF) satisfy?

Answer 34

A continuous random variable XX is defined by a nonnegative PDF f(x)f(x), where P(X∈B)=∫Bf(x)dxP(X∈B)=∫B​f(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.

Question 35

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)?

Answer 35

Solve ∫06C(6x−x2)dx=1∫06​C(6x−x2)dx=1. Integration yields C[3x2−x33]06=36C=1C[3x2−3x3​]06​=36C=1, so C=136C=361​.

Question 36

For f(x)=136(6x−x2)f(x)=361​(6x−x2), what is the probability X>3X>3?

Answer 36

Compute P(X>3)=∫36136(6x−x2)dxP(X>3)=∫36​361​(6x−x2)dx. Result: 136[3x2−x33]36=12361​[3x2−3x3​]36​=21​.

Question 37

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?

Answer 37

Probability of failure within 150h: ∫100150100x2dx=13∫100150​x2100​dx=31​. For 2/5 failures: Use the binomial formula: (52)(13)2(23)3≈0.329(25​)(31​)2(32​)3≈0.329.

Question 38

How does scaling a continuous RV XX to Y=5XY=5X affect its PDF?

Answer 38

If XX has PDF fX(x)fX​(x), then Y=5XY=5X has PDF fY(y)=15fX(y5)fY​(y)=51​fX​(5y​). This follows from transforming the CDF: FY(y)=FX(y5)FY​(y)=FX​(5y​).

Question 39

Calculate E[X]E[X] for f(x)=3x2f(x)=3x2 on [0,1][0,1].

Answer 39

E[X]=∫01x⋅3x2dx=3∫01x3dx=3⋅14=34E[X]=∫01​x⋅3x2dx=3∫01​x3dx=3⋅41​=43​.

Question 40

Derive the variance for f(x)=3x2f(x)=3x2 on [0,1][0,1].

Answer 40

First compute E[X2]=∫01x2⋅3x2dx=35E[X2]=∫01​x2⋅3x2dx=53​. Then Var(X)=E[X2]−(E[X])2=35−(34)2=380Var(X)=E[X2]−(E[X])2=53​−(43​)2=803​.

Question 41

If Var(X)=3.5Var(X)=3.5, what is Var(2X−7)Var(2X−7)?

Answer 41

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.

Question 42

Uniform RV X∼(3,8)X∼(3,8): What is P(X<5)P(X<5)?

Answer 42

For uniform (α,β)(α,β), P(a

Question 43

What are the expectation and variance formulas for a uniform RV (α,β)(α,β)?

Answer 43

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​.

Question 44

Passengers arrive uniformly between 7:00-7:30. What’s the probability of waiting less than 5 minutes for a bus?

Answer 44

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​.

Question 45

Why does E[X1+⋯+Xn]=E[X1]+⋯+E[Xn]E[X1​+⋯+Xn​]=E[X1​]+⋯+E[Xn​], even for dependent variables?

Answer 45

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.

Question 46

How is the expected number of white balls selected in a hypergeometric experiment derived?

Answer 46

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=1m​E[Xi​]=m⋅Nn​, since P(Xi=1)=nNP(Xi​=1)=Nn​.

Question 47

In the hat-matching problem, why is the expected number of matches 1?

Answer 47

Let Ii=1Ii​=1 if person ii gets their hat. E[Ii]=1NE[Ii​]=N1​, so E[X]=∑i=1N1N=1E[X]=∑i=1N​N1​=1. Linearity simplifies complex dependencies.

Question 48

What does covariance measure, and how is it calculated?

Answer 48

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.

Question 49

How is the variance of a sum of RVs affected by covariance?

Answer 49

Var(X1+⋯+Xn)=∑Var(Xi)+2∑i

Question 50

If Y=a+bXY=a+bX, why is ρ(X,Y)=1ρ(X,Y)=1 for b>0b>0?

Answer 50

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.

Question 51

For 10 fair dice rolls, how is the total variance calculated?

Answer 51

Single die variance: Var(Xi)=3512Var(Xi​)=1235​. Total variance: 10×3512=1756≈29.1710×1235​=6175​≈29.17. Independence ensures variances add directly.