[Maximum number: 11]

A random variable X has probability density function

f(x)=\left\{\begin{array}{cc} 0 & x<0 \\ \frac{1}{2} & 0 \leq x<1 \\ \frac{1}{4} & 1 \leq x<3 \\ 0 & x \geq 3 \end{array} .\right.

(a)

Sketch the graph of y=f(x).

[ 1 ]

(b)

Find the cumulative distribution function for X.

[ 5 ]

(c)

Find the interquartile range for X.

[ 5 ]

[Maximum number: 5]

A continuous random variable X has the probability density function f given by

f(x)=\left\{\begin{array}{cl} c\left(x-x^{2}\right), & 0 \leq x \leq 1 \\ 0, & \text { otherwise } \end{array}\right.

(a)

Determine c.

[ 3 ]

(b)

Find E(X).

[ 2 ]

(a)

A hospital specializes in treating overweight patients. These patients have weights that are independently, normally distributed with mean 200 kg and standard deviation 15 kg . The elevator in the hospital will break if the total weight of people inside it exceeds 1150 kg . Six patients enter the elevator. Find the probability that the elevator breaks.

[ 7 ]

(b)

A factory makes life size wax copies of famous people. These famous people have weights that are independently, normally distributed with mean 80 kg and standard deviation 10 kg . The life size copies all have exactly the same weight as the famous person they represent. Twelve copies of one particular famous person are placed in the elevator in the factory. This elevator will also break if the total weight of the copies exceeds 1150 kg . Find the probability that the elevator breaks.

[ 7 ]

[Maximum number: 5]

The random variable X has probability distribution $\operatorname{Po}(8)$ .

(a)

$\bar{X}$ denotes the sample mean of n>1 independent observations from X.

[ 5 ]

(i)

Write down $\mathrm{E}(\bar{X})$ and $\operatorname{Var}(\bar{X})$ .

(ii)

Hence, give a reason why $\bar{X}$ is not a Poisson distribution.
(b) $\bar{X}$ denotes the sample mean of n>1 independent observations from X.
(i) Write down $\mathrm{E}(\bar{X})$ and $\operatorname{Var}(\bar{X})$ .

Hence, give a reason why $\bar{X}$ is not a Poisson distribution.

[ 5 ]

[Maximum number: 5]

A random variable X has a probability distribution given in the following table.

(a)

Determine the value of $\mathrm{E}\left(X^{2}\right)$ .

[ 2 ]

(b)

Find the value of $\operatorname{Var}(X)$ .

[ 3 ]

[Maximum number: 7]

A continuous random variable T has a probability density function defined by

f(t)=\left\{\begin{array}{cc} \frac{t\left(4-t^{2}\right)}{4}, & 0 \leq t \leq 2 \\ 0, & \text { otherwise } \end{array}\right.

(a)

Find the cumulative distribution function F(t), for $0 \leq t \leq 2$ .

[ 3 ]

(b)

(i)

Sketch the graph of F(t) for $0 \leq t \leq 2$ , clearly indicating the coordinates of the endpoints.

(ii)

Given that P(T<a)=0.75, find the value of a.

[ 4 ]

[Maximum number: 16]

The continuous random variable X has a probability density function given by

f(x)=\left\{\begin{array}{cc} k x & 0 \leq x<1 \\ k x^{2} & 1 \leq x \leq 2 \\ 0 & \text { otherwise } \end{array}\right.

(a)

Show that $k=\frac{6}{17}$ .

[ 4 ]

(b)

Find the cumulative distribution function of X.

[ 6 ]

(c)

Find the median, m, of X.

[ 3 ]

(d)

Find $\mathrm{P}(|X-m|<0.75)$ .

[ 3 ]

[Maximum number: 4]

The probability density function of the continuous random variable X is given by

f(x)=\left\{\begin{array}{cl} k 2^{\frac{1}{x}}, & 1 \leq x \leq 2 \\ 0, & \text { otherwise } \end{array}\right.

where k is a constant. Find the expected value of X.

[Maximum number: 6]

3.
In a particular city 20 % of the inhabitants have been immunized against a certain disease. The probability of infection from the disease among those immunized is $\frac{1}{10}$ , and among those not immunized the probability is $\frac{3}{4}$ . If a person is chosen at random and found to be infected, find the probability that this person has been immunized.

[Maximum number: 5]

4.
Jenny goes to school by bus every day. When it is not raining, the probability that the bus is late is $\frac{3}{20}$ . When it is raining, the probability that the bus is late is $\frac{7}{20}$ .
The probability that it rains on a particular day is $\frac{9}{20}$ . On one particular day the bus is late. Find the probability that it is not raining on that day.

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