These measures can be calculated using
i. Mathematical Expectation and variance.
ii. Moment Generating Function .
iii. Characteristic Function.

**Mathematical
expectation**

Probability
distribution gives us an idea about the likely value of a random variable and
the probability of the various events related to random variable. Even though
it is necessary for us to explain probabilities using central tendencies,
dispersion, symmetry and kurtosis. These are called descriptive measures and
summary measures. Like frequency distribution we have to see the properties of
probability distribution. This section focuses on how to calculate these
summary measures. These measures can be calculated using

i. Mathematical Expectation and variance.

ii. Moment Generating Function .

iii. Characteristic Function.

Let *X* be a discrete
random variable which takes the values *x*_{1},
*x*_{2} ….. *x _{n}* with respective probabilities

Sometimes *E*(*X*) is known as the mean of the random
vairable *X*.

If g(X) is
a function of the random variable X, then E g(X) = Σg(x)p(X = x)

*E*(*c*) =* c *where* c *is a constant

**Proof:**

Definition: In a probability distribution Variance is the
average of sum of squares of deviations from the mean. The variance of the
random variable X can be defined as.

Var ( *X* ) = *E* (*X*
– *E*(*X*))^{2}

= *E* (*X*^{2} ) – ( *E*(*X*))^{2}

When a die is thrown *X*
denotes the number turns up. Find *E*(*X*), *E*(*X*^{2}) and Var(*X*).

Let *X* denoted that
number turns up in a die.

X takes the values 1, 2, 3, 4, 5, 6 with probabilities 1/6 for
each.

Therefore the probability distribution is

The
mean and standard deviation of a random variable *X* are 5 and 4 respectively Find *E*
(X^{2})

Given
E (X) = 5 and σ = 4

`
Var (X) = 16

But,
Var (X) = E(X^{2}) – [E(X)]^{2}

16
= E (X^{2}) – (5)^{2}

E(X^{2})
= 25 + 16 = 41

A
player tosses two coins, if two head appears he wins ` 4, if one head appears
he wins ` 2, but if two tails appears he loses ` 3. Find the expected sum of
money he wins?

Let
*X* be the random variable denoted the amount he wins.

The
possible values of *X* are 4, 2 and –3

Let *X* be a discrete
random variable with the following probability distribution

Find mean and variance

Mean = *E*(*X*) = ∑*x*_{i} *p*_{i}

Let *X* be a continuous
random variable with probability density function *f* (x) then the mathematical expectation of X is defined as

*E*(*c*) = *c *where c is constant

The
p.d.f. of a continuous random variable *X* is given by

find its mean and variance

So
far we have studied how to find mean and variance in the case of single random
variables taken at a time but in some cases we need to calculate the
expectation for a linear combination of random variables like *aX*
+ *bY* or the product of the random variables *cX *×* dY *or
involving more number of random variables. So here we see theorems useful* *in
such situations.

Random
variables *X* and *Y* are said to be independent if the
joint probability density function of X and Y can be written as the product of
marginal densities of *X* and *Y*.

That
is *f* (*x*,*y*)= *g*(*x*) *h*(*y*)

Here *g*(*x*) marginal p.d.f. of* X*

*h*(*y*) marginal p.d.f. of* Y*

Tags : Result, Properties, Proof, Solved Example Problems , 11th Statistics : Chapter 9 : Random Variables and Mathematical Expectation

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11th Statistics : Chapter 9 : Random Variables and Mathematical Expectation : Mathematical expectation | Result, Properties, Proof, Solved Example Problems

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