The document discusses expected value, which is an anticipated average value for an investment or random variable at some point in the future. Expected value is also called the mean. It can be calculated as the sum of the products of each possible value and its probability for discrete random variables, and as an integral of the random variable multiplied by its probability density function for continuous random variables. As an example, the document calculates the expected or mean time it takes to re-heat a cup of coffee based on its probability density function.

1. 4.2 EXPECTED VALUES OF
CONTINUOUS RANDOM
VARIABLES
The expected value (EV) is an anticipated average value for an
investment at some point in the future. Investors use expected value
to estimate the worthiness of investments, often in relation to their
relative riskiness.

2. Expected value or Mathematical Expectation or Expectation of a random
variable may be defined as the sum of products of the different values
taken by the random variable and the corresponding probabilities.
E ( X ) = μ = ∑ x P ( x )
Let "x" be a continuous random variable which is defined in the interval (-∞ , +∞),
with the probability density function f(x).

3. Then, the expected value is given by
Expected value of x² is given by

4. NOTE:
Expected value is also called as mean.

5. PROPERTIES OF
EXPECTED
VALUE
0
1
02
Expectation of a
constant k is k
Expectation of sum of
two random
variables is the sum
of their expectations.
That is,
E(k) = k for any
constant k
That is,
E(x + y) = E(x) + E(y) for
any two random
variables x and y.

6. EXPECTED
VALUE
0
3
0
4
Expectation of the product
of a constant and a
random variable is the
product of the constant
and the expectation of the
random variable.
Expectation of the product
of two random variables is
the product of the
expectation of the two
random variables,
provided the two variables
are independent.
That is,
E(kx) = k.E(x) for any constant
k
That is,
E(xy) = E(x).E(y)

7. FOR
EXAMPLE
THE TIME, IN SECONDS, IT TAKES TO RE-HEAT A CUP OF COFFEE
CAN BE MODELLED BY THE CONTINUOUS RANDOM VARIABLE X,
WITH PROBABILITY DENSITY FUNCTION:
f (x)={⅜ x², 0 ≤ x ≤ 2 0, elsewhere
Calculate the mean amount of time it takes to re-heat a cup of coffee.