Continuous random variables

A continuous random variable X has probability density function

where a and b are constants.

use algebraic integration to show that a = 3 Show your working clearly.

Find the exact value of the median of X

The continuous random variable X has cumulative distribution function given by

The values of a and k are positive constants such that P(X < 2) =

Find the exact value of the median of X

Find the probability density function of X

Hence, deduce the value of the mode of X, giving a reason for your answer.

The continuous random variable X has cumulative distribution function given by

Find the exact value of the median of X

Find P(X 1.6 | X 1.2)

Specify fully the cumulative distribution function of Y

Hence or otherwise find the mode of Y

The continuous random variable X has cumulative distribution function given by

Find P(X < 5X 8)

Find the median of X.

Find E(X²)

Sketch the probability density function of X.

Describe the skewness of the distribution of X.

The random variable X has probability density function f (x) and Figure 1 shows a sketch of f (x) where

The random variable Y~ N (μ, σ²) and E(Y) = E(X)

The probability density function of Y is g (y), where

Given that g (μ) = f (μ)

find the exact value of σ

The continuous random variable has cumulative distribution function given by

where k is a constant.

Showing your working clearly, use calculus to find

E(X)

the mode of X

Describe, giving a reason, the skewness of the distribution of X