(a)

If is a probability density function then it must satisfy .

Observe the graph:

The graph of is a triangular region.

Find .

Calculate the area under the graph of with from to .

The triangle height is units and base is units.

Area of the triangle .

.

and for all .

Therefore, the function whose graph is shown is a probability density function.

(b)

(i)

Find .

.

.

From the graph, .

Find .

Calculate the area under the graph of with from to .

The triangle height is units and base is units.

.

.

(b)

(ii)

Find .

.

Calculate the area under the graph of with from to .

From part (b)(i) .

.

Estimate the area under the graph of with from to .

The triangle height is units and base is units.

.

.

.

(c)

The mean of any probability density function is defined to be .

.

Observe the graph:

Find .

if .

.

Find .

if .

.

The mean is .

.

(a), whose graph is shown is a probability density function.

(b) ; .

(c) .