Discrete Probability Distributions - Uniform Distribution

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What is the formula for the mean of a binomial distribution with parameters n and p?

E[X] = n(1 - p)

E[X] = nq

E[X] = n/p

E[X] = np (correct)

Which distribution describes the number of events occurring in a fixed interval of time or space with a known constant rate?

Binomial distribution

Normal distribution

Uniform distribution

Poisson distribution (correct)

What is the variance formula for a Poisson distribution with parameter λ?

$Var(X) = λ$ (correct)

$Var(X) = λ^2$

$Var(X) = rac{1}{λ^2}$

$Var(X) = rac{1}{λ}$

In the moment generating function formula for binomial distribution, what is represented by 't'?

Variable for transformation Signup and view all the answers

For which distribution does the probability mass function contain the term k!?

Poisson distribution Signup and view all the answers

What does the moment generating function help calculate for a probability distribution?

Mean, variance, and higher moments Signup and view all the answers

In a continuous uniform distribution, every value between a and b has what kind of likelihood of occurring?

Equal likelihood Signup and view all the answers

Which formula correctly represents the moment generating function for a Poisson distribution with parameter λ?

!MX?(t) = e^λ(et?1) Signup and view all the answers

What is the variance formula for a binomial distribution B(n,p)?

$Var(X) = np(1-p)$ Signup and view all the answers

What is the notation for the uniform distribution with minimum value $a$ and maximum value $b$?

U(a,b) Signup and view all the answers

For the uniform distribution U(a,b), what is the probability mass function (PMF) when $x=a+3$?

$P(X=a+3) = 0$ Signup and view all the answers

What is the variance of a uniform distribution U(2,8)?

$4.5$ Signup and view all the answers

What does the moment generating function (MGF) help derive for a random variable?

Moments of the Distribution Signup and view all the answers

What is the moment generating function (MGF) for a uniform distribution over integers 1 to 6?

$(6e^t-1)e^{7t}$ Signup and view all the answers

In the context of a uniform distribution U(a,b), if $P(X=x)$ for $x=a+2$, what is $P(X=a+2)$?

$\frac{1}{b-a+1}$ Signup and view all the answers

What is the mean of a uniform distribution U(3,9)?

$5.5$ Signup and view all the answers

If a uniform distribution U(4,10) has $P(X=x) = \frac{1}{7}$ for some $x$, what does this suggest about $x$?

$x=10$ Signup and view all the answers

For a uniform distribution U(2,7), what is the probability of $P(X=3)$?

$0.1667$ Signup and view all the answers

What does the moment generating function MX(t) help calculate for a random variable X?

$E[X]$ Signup and view all the answers