What type of random variable takes on a countable number of distinct values?

Discrete random variable

True or False: The probability density function gives exact probabilities for continuous variables.

False. PDF gives relative likelihood; probabilities require integration over intervals.

For a binomial distribution with n=10 and p=0.3, what is μ?

Μ = np = 10 × 0.3 = 3

What distribution models time between events in a Poisson process?
A) Normal
B) Exponential
C) Uniform
D) Binomial

B) Exponential

The _____ property means P(X > s+t | X > s) = P(X > t).

Memoryless

What does the Central Limit Theorem state about sample means?

Sample means approach normal distribution for large n, regardless of original distribution.

For Poisson distribution with λ = 5, what are μ and σ²?

Μ = λ = 5 and σ² = λ = 5

True or False: In standard normal distribution, μ = 1 and σ = 0.

False. Standard normal has μ = 0 and σ = 1.

A quality inspector finds 2% defect rate. For 50 items, which distribution applies?
A) Normal
B) Exponential
C) Binomial
D) Uniform

C) Binomial

What percentage of normal distribution values lie within μ ± 2σ?

Approximately 95%

For uniform distribution on [2,8], what is the mean?

Μ = (a+b)/2 = (2+8)/2 = 5

The formula Z = (X-μ)/σ is used for what process?

Standardization (converting to standard normal distribution)

True or False: CDF gives P(X ≤ x) for any random variable.

True

What happens to Var(X̄) as sample size n increases?

Var(X̄) = σ²/n decreases as n increases

Events occur at rate λ=3 per hour. What's P(X=0) in one hour?
A) e⁻³
B) 3e⁻³
C) 1-e⁻³
D) 3

A) e⁻³

For exponential distribution with λ=0.5, what is the mean time?

Μ = 1/λ = 1/0.5 = 2

The _____ function describes relative likelihood for continuous random variables.

Probability density function (PDF)

True or False: Binomial can approximate Poisson when n is small and p is large.

False. Binomial approximates Poisson when n is large and p is small.

What is the variance formula using expectation notation?

Var(X) = E(X²) - [E(X)]²

Machine failures follow exponential distribution, mean = 100 hours. After 50 hours, what's probability of 25 more hours?
A) e⁻⁰·²⁵
B) e⁻⁰·⁵
C) e⁻⁰·⁷⁵
D) e⁻¹

A) e⁻⁰·²⁵

For normal distribution, what does the 68-95-99.7 rule describe?

Percentages of data within 1σ, 2σ, and 3σ of the mean

What is the binomial coefficient formula for C(n,x)?

C(n,x) = n!/(x!(n-x)!)

True or False: PDF must satisfy ∫f(x)dx = 1 over entire domain.

True

When is Central Limit Theorem typically considered applicable?

When sample size n ≥ 30

For uniform distribution on [a,b], what is the variance?

Σ² = (b-a)²/12

Phone calls arrive with λ=2 per minute. What's expected calls in 30 minutes?
A) 2
B) 30
C) 60
D) 32

C) 60