Continuous uniform distribution

The random variable G has a continuous uniform distribution over the interval [– 3, 15]

Calculate P(G 12)

The random variable H has a continuous uniform distribution over the interval [2, w]

The random variables G and H are independent and E(H) = 10

Show that the probability that G and H are both greater than 12 is

The random variable A is the area on a coordinate grid bounded by

y = –3

y = –4 | x | + k

where k is a value from the continuous uniform distribution over the interval [5, 10]

Calculate the expected value of A

The random variable R has a continuous uniform distribution over the interval [2, 10]

Write down the probability density function f(r) of R

A sphere of radius R cm is formed.

The surface area of the sphere, S cm², is given by S = 4πR²

The volume of the sphere, V cm, is given by V ϖ R³

Find, using algebraic integration, the expected value of V

A rectangle is to have an area of 40 cm²

The length of the rectangle, Lcm, follows a continuous uniform distribution over the interval [4, 10]

Find the expected value of the perimeter of the rectangle. Use algebraic integration, rather than your calculator, to evaluate any definite integrals.

The continuous random variable X is uniformly distributed over the interval [0, 4β], where β is an unknown constant.

Three independent observations, X₁, X₂ and X₃, are taken of X and the following estimators for β are proposed

Calculate the bias of A, the bias of B and the bias of C

By calculating the variances, explain which of B or C is the better estimator for β

Find an unbiased estimator for β

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.