Measures of location

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.

A circle, centre O, has radius xcm, where x is an observation from the random variable X which has a rectangular distribution on [0, π ]

Find the probability that the area of the circle is greater than 10 cm²

State, giving a reason, whether the median area of the circle is greater or less than

The triangle OAB is drawn inside the circle with OA and OB as radii of length xcm and angle AOB x radians.

Use algebraic integration to find the expected value of the area of triangle OAB. Give your answer as an exact value.

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