QTA 3 - COMMON UNIVARIATE RANDOM VARIABLES Flashcards
What are the key properties among the following distributions: Uniform, Bernoulli, Poisson, Normal, Lognormal, Chi-squared, Studentβs t, and F?
They have distinct characteristics and applications in modeling different types of data.
Each distribution is utilized in specific contexts, such as modeling binary events, counts of events, or continuous data.
What is a Bernoulli distribution?
A discrete distribution for random variables that produces one of two values: 0 or 1.
It models binary outcomes like success/failure.
What parameter does the Bernoulli distribution depend on?
The probability of success, denoted as p.
The distribution is expressed as Y ~ Bernoulli(p).
What is the mean of a Bernoulli random variable Y?
E[Y] = p.
This is calculated as p * 1 + (1 - p) * 0.
What is the variance of a Bernoulli random variable Y?
V[Y] = p(1 - p).
Where q = 1 - p is the failure probability.
What is the probability mass function (PMF) of a Bernoulli distribution?
f(y) = p^y * (1 - p)^(1 - y).
This function only produces two values: p when y = 1 and (1 - p) when y = 0.
What is a binomial distribution?
It describes the sum of n independent Bernoulli random variables.
It models the total number of successes in n trials.
What parameters define a binomial distribution?
n (number of trials) and p (probability of success).
The binomial distribution is expressed as Y = B(n, p).
What is the mean of a binomially distributed random variable Y?
E[Y] = n * p.
This represents the expected number of successes in n trials.
What is the variance of a binomially distributed random variable Y?
V[Y] = n * p * (1 - p).
This accounts for the variability in the number of successes.
What is the skewness of a binomial distribution dependent on?
The probability p; small values produce right-skewed distributions.
This affects how the distribution is shaped.
What is a Poisson distribution used for?
To model counts of events over fixed time spans.
Examples include loan defaults or customer arrivals.
What is the single parameter of a Poisson distribution?
The hazard rate, denoted as Ξ».
This represents the average number of events per interval.
What are the mean and variance of a Poisson random variable Y?
Both are equal to Ξ».
This property simplifies calculations for Poisson processes.
What is the PMF of a Poisson random variable?
P(Y = n) = (Ξ»^n * e^(-Ξ»)) / n!.
This defines the probability of observing n events.
What is a uniform distribution?
A distribution where any value within the range [a, b] is equally likely to occur.
It is the simplest continuous random variable.
What is the PDF of a uniform distribution?
f(y) = 1 / (b - a) for a β€ y β€ b.
It is zero outside this range.
What is the mean of a uniform random variable Y ~ U(a, b)?
E[Y] = (a + b) / 2.
This represents the midpoint of the distributionβs support.
What is the variance of a uniform random variable Y ~ U(a, b)?
V[Y] = (b - a)^2 / 12.
This shows the distributionβs spread.
What is the normal distribution often referred to as?
Gaussian distribution or bell curve.
It is widely used in risk management.
What does the normal distribution play a key role in?
The Central Limit Theorem (CLT).
This is crucial for hypothesis testing.
What are the mean and variance of a normal distribution Y ~ N(ΞΌ, ΟΒ²)?
E[Y] = ΞΌ and V[Y] = ΟΒ².
These parameters fully describe the distribution.
What is the confidence interval for a 95% confidence level in a normal distribution?
ΞΌ - 1.96Ο to ΞΌ + 1.96Ο.
This interval contains approximately 95% of the data.
True or False: The normal distribution is infinitely divisible.
True.
This property allows for flexible modeling of random processes.
What is the formula for a 90% confidence interval?
π β 1.645π to π + 1.645π
What is the formula for a 95% confidence interval?
π β 1.96π to π + 1.96π
What is the formula for a 99% confidence interval?
π β 2.58π to π + 2.58π
What is a standard normal distribution?
A normal distribution standardized to have a mean of zero and a standard deviation of one, denoted as π~π(0,1)
What does Ρ(π§) denote?
The standard normal PDF
What does Ξ¦(π§) denote?
The standard normal CDF
What is the formula for converting an observed value π¦ to its z value?
z = (π¦ β π) / π
What are the conditions for a normal distribution to approximate a binomial random variable?
- ππ β₯ 10
- π(1 β π) β₯ 10
When can a Poisson distribution be approximated by a normal distribution?
When π is large, specifically when π β₯ 1000
What defines a log-normally distributed variable?
A variable π is log-normally distributed if the natural logarithm of π is normally distributed
What is the relationship between π and π in a log-normal distribution?
If π = ππ(π), then π is log-normally distributed if and only if π is normally distributed
What is the mean of a log-normally distributed variable π?
E[π] = e^(ΞΌ + 1/2Ο^2)
What is the variance of a log-normally distributed variable π?
V[π] = e^(2ΞΌ + Ο^2) - e^(2ΞΌ + Ο^2)
What is the characteristic of a log-normal distribution?
It is positively skewed
What is the PDF of a log-normal distribution?
π(π¦) = (1 / (π¦Οβ(2Ο))) * e^(-(ln(π¦) - ΞΌ)^2 / (2Ο^2))
What is the chi-squared distribution frequently used for?
Testing hypotheses about model parameters
How is a chi-squared random variable defined?
As the sum of the squares of π independent standard normal random variables
What is the mean of a chi-squared distribution?
E[π] = π
What is the variance of a chi-squared distribution?
V[π] = 2π
What is the Studentβs t distribution used for?
Testing hypotheses using small samples
What is the mean of a Studentβs t distribution?
E[π] = 0
What is the variance of a Studentβs t distribution?
V[π] = π / (π - 2), finite if π > 2
What is the F distribution commonly used for?
Testing hypotheses about model parameters
What is the mean of an F distribution?
E[π] = π2 / (π2 - 2), finite when π2 > 2
What is the variance of an F distribution?
V[π] = (2π2(π1 + π2 - 2)) / (π1(π2 - 2)^2), finite for π2 > 4
What is the exponential distribution used for?
Modeling time until an event occurs, with a single parameter π½
What is the mean of an exponentially distributed variable?
E[π] = π½
What is the variance of an exponentially distributed variable?
V[π] = π½^2
What does the beta distribution model?
Continuous random variables with outcomes between 0 and 1
What are the parameters of a beta distribution?
Ξ± and π½
What is the mean of a beta-distributed random variable?
E[π] = Ξ± / (Ξ± + π½)
What is the variance of a beta-distributed random variable?
V[π] = (Ξ±π½) / ((Ξ± + π½)^2(Ξ± + π½ + 1))
What are mixture distributions?
Distributions built using two or more component distributions
What is a characteristic of mixing components with different means and variances?
Produces a distribution that is both skewed and heavy-tailed
