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Discrete Probability Distributions - Uniform Distribution

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@SprightlyVision

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'?

<p>Variable for transformation</p> Signup and view all the answers

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

<p>Poisson distribution</p> Signup and view all the answers

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

<p>Mean, variance, and higher moments</p> Signup and view all the answers

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

<p>Equal likelihood</p> Signup and view all the answers

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

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

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

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

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

<p>U(a,b)</p> 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>$P(X=a+3) = 0$</p> Signup and view all the answers

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

<p>$4.5$</p> Signup and view all the answers

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

<p>Moments of the Distribution</p> Signup and view all the answers

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

<p>$(6e^t-1)e^{7t}$</p> 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 $P(X=a+2)$?

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

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

<p>$5.5$</p> 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 $x$?

<p>$x=10$</p> Signup and view all the answers

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

<p>$0.1667$</p> Signup and view all the answers

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

<p>$E[X]$</p> Signup and view all the answers

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