# Discrete Random Variables, normal distribution

**Discrete Random Variables, normal distribution**

- Created by: Rukiya Majid
- Created on: 13-05-09 14:27

First 317 words of the document:

S1 Revision Notes: Discrete Random Variables

Recap: In a statistical experiment the outcome is unknown in advance. The outcome of the experiment is

called a random variable. If the experiment can result in only a fixed number of outcomes, it is called a

discrete random variable.

The outcomes of the experiment and the probabilities form a probability distribution. They can be shown in

a table:

A discrete random variable has certain properties:

The sum of all the probabilities is 1, i.e.

Each probability is between 0 and 1, i.e.

There are 2 key formulae:

E(X) = (i.e. times the top and bottom rows together and add).

Var(X) = E(X²) ² where = E(X)

and E(X²) =

Example:

A shop sells gift vouchers valued at £1, £2, £4, £10 or £20. The value in £ of a gift voucher sold may be

regarded as a random variable, X, with the following distribution:

x 1 2 4 10 20

P(X = x) 0.20 0.40 0.22 0.11 0.07

a) Find the mean and the standard deviation of X.

b) What is the probability that the value of the next voucher sold is less than £4.

The shop is considering whether to discontinue selling £1 and £2 vouchers. If they did this a proportion p

of customers who presently buy £1 and £2 vouchers would then buy a £4 voucher. Other such customers

would not buy a voucher. As a result the value, in £, of sales would be a random variable Y with the

following distribution.

y 0 4 10 20

P(Y = y) 0.6(1 p) 0.22 + 0.6p 0.11 0.07

c) Find the mean of Y in terms of p.

d) A survey suggests that the value of p would be between 0.5 and 0.7. Using this information,

advise the shop on the likely effect on takings if she decides to discontinue selling £1 and £2

vouchers.

1

Created on 12/05/2005 10:12 AM by A. Duncombe

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