Expectation & variance

The proportion X of charge in a phone battery is measured on a scale from 0 (uncharged) to 1 (fully charged).

For a certain model of phone, X can be modelled as a continuous random variable with the following probability density function.

where k is a positive constant.

Show that f(x) is a valid probability density function for all positive values of k.

Suppose that k = 3.

Determine Var(X).

Suppose instead it is known that X is less than 0.8203 for 50% of this model of phone.

Determine the 70th percentile of X.

A discrete random variable D has the following probability distribution, where a is a constant.

Determine the value of Var(3 D + 4).

The continuous random variable Y has a uniform distribution on [0, 2].

It is given that E[ a cos(aY)] = 0.3, where a is a constant between 0 and 1, and aY is measured in radians.

Determine the value of the constant a.

The continuous random variable X has probability density function

where n and k are constants and n is an integer greater than 1.

Find k in terms of n.

When n = 4, find the cumulative distribution function of X.

Hence determine P (X > 7 | X > 5) when n = 4.

Determine the values of n for which Var(X) is not defined.