Continuous Distributions 2 (AQA A-Level Further Maths): Quizzes

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Continuous Distributions 2
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For a continuous random variable XX, what is P(X=5)P(X = 5)?

00

How do you find the PDF f(x)f(x) from the CDF F(x)F(x)?

Differentiate: f(x)=ddxF(x)f(x) = \frac{d}{dx}F(x)

What condition must a valid PDF f(x)f(x) satisfy over its domain [a,b][a, b]?

∫abf(x)dx=1\int_a^b f(x)dx = 1

The median MM of a continuous distribution satisfies which equation?

F(M)=0.5F(M) = 0.5

If F(x)=1−e−λxF(x) = 1 - e^{-\lambda x} for x≥0x \geq 0, what is the PDF f(x)f(x)?

λe−λx\lambda e^{-\lambda x}

For the exponential PDF f(x)=λe−λxf(x) = \lambda e^{-\lambda x} where x≥0x \geq 0, what is the mode?

00

For R∼Rectangular(a,b)R \sim \text{Rectangular}(a, b), what is the PDF f(r)f(r) for a<r<ba < r < b?

1b−a\frac{1}{b-a}

What is the mean of a rectangular distribution on (a,b)(a, b)?

a+b2\frac{a+b}{2}

What is the variance of R∼Rectangular(a,b)R \sim \text{Rectangular}(a, b)?

(b−a)212\frac{(b-a)^2}{12}

For a mixed distribution with discrete probability cc and continuous PDF f(x)f(x), what must equal 1?

c+∫f(x)dxc + \int f(x)dx

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