# Uniform Random Variables and Uniform Distribution

## Continuous Uniform Random Variables:

Continuous Uniform Random Variables are used to model a scenario where a continuous random variable can take values that are equally distributed (with equal probability) in an interval. Uniform distribution is the underlying distribution to model such random variables.

A continuous uniform distribution is denoted by

$$X \sim \mathcal{U}(a,b) \text{, with a} \leq b $$

Here \( a\) and \(b\), are limits of the interval over which the distribution is effectively defined.

### Probability Density Function

The Probability Density Function (PDF) of the continuous uniform random distribution defined above is given by

$$ f_X(x) = \begin{cases} \frac{1}{b-a} & \text{ if } a \leq x \leq b \\ 0 & \text{ otherwise} \end{cases}$$

This equation indicates that the distribution when used, generates values in the interval (a,b) with equal probability (with probability \( \frac{1}{b-a}\))

### Continuous Density Function

The Continuous Density Function (CDF) can be derived from the above PDF using the relation

$$ F_X(x) = \int_{-\infty }^{x}f_X(t)dt$$

Therefore, the CDF of the continuous uniform distribution is given by,

$$F_X(x) = \begin{cases} 0 & \text{ if } x < a \\ \frac{x-a}{b-a} & \text{ if } x \in \left[ a,b \right) \\ 1 & \text{ if } x \geq b \end{cases} $$

Plotting the PDF and CDF,

## Discrete Uniform Random Variable:

Discrete Uniform Random Variables are used to model a scenario where a discrete random variable can take values that are equally distributed (with equal probability) in an interval. The underlying discrete uniform distribution is denoted by

$$ X \sim \mathcal{U}(\mathcal{S}) $$

Where\( \mathcal{S} \) is a finite set of discrete points on the real line. Meaning that each element in the finite set are equally likely.

### Example:

“Rolling a fair die” can be considered as an experiment to generate uniform integers in the range 1 to 6. Since rolling of a fair die give only one of the six possible integer values (1,2,3,4,5,6), it is an experiment that generates discrete values. Since all the values in a fair die are equally likely (with probability of getting each value =1/6) it can be modelled using a discrete uniform distribution.

For a die, the set \( \mathcal{S} \) can take values \(\{1,2,3,4,5,6\}\)

### Probability Mass Function (PMF):

If \( \mathcal{S} = \left\{ s_1, \cdots ,s_n \right\} \in \mathcal{Z} \) , then the due to the definition of uniform distribution, each element in the set are equally probable. Therefore,

\( Prob \left[X=s_1 \right]=Prob \left[X=s_2 \right]=\cdots=Prob \left[X=s_n \right]=\frac{1}{n}\)

Then the PMF can be given by,

$$ f_X(x) = \begin{cases} \frac{1}{n} & x \in \left\{s_1,s_2 \cdots , s_n \right\} \\ 0 & otherwise \end{cases}$$

For rolling a fair die, the PMF simplifies as follows

$$ f_X(x) = \begin{cases} \frac{1}{6} & x \in \left\{1,2,3,4,5,6 \right\} \\ 0 & otherwise \end{cases}$$

The PMF and CDF for a discrete uniform random distribution is shown below

## Generating Uniform Random Numbers in Matlab:

### Generating Continuous Uniform Random Numbers:

In Matlab **“rand”** function is used to generate continous uniform random numbers in the range [0,1].

rand function is used to pick any number in the interval [0,1] in which all the numbers in the interval are equally likely.

**rand(n,m)** will generate a matrix of size nxm. A trial run at my system gave the following results

1 2 3 4 5 6 |
>>rand(3,4) >> ans = 0.2920 0.3900 0.4332 0.4159 0.0948 0.6802 0.8264 0.7536 0.6770 0.8449 0.6083 0.7715 |

To generate a random number in the interval [a,b] one can use the following code

** a + (b-a)*rand(n,m); %Here nxm is the size of the output matrix **

For example, to generate a matrix of size \(3 \times 4\) using uniform distribution in the interval [2,10]

1 2 3 4 5 6 |
>> 2 + (10-2)*rand(3,4) >> ans= 5.4852 4.5016 9.6905 2.0585 2.7066 8.4537 9.8910 4.4816 3.1029 5.2562 6.9187 8.7710 |

### Testing Uniformity:

To test whether the numbers generated by the uniform distribution are uniform in the interval [a,b], one has to generate very large number of values using the rand function and then plot the histogram

Testing the uniformity of **rand** function in the interval [2,10]

1 2 |
>> x = 2+(10-2)*rand(1,100000); >> hist(x); |

The histogram plot given below shows that the numbers in the range [2,10] are generated with almost equal probability.

### Generating Discrete Uniform Random Numbers:

The same “rand” function can be used to generate discrete random numbers that follow discrete uniform distribution. The code to generate uniformly distributed discrete numbers from the set \( \mathcal{S}=\left\{1,2,3,…,n\right\}\) is given below.

1 |
r = ceil(n.*rand(1,100)); %This will generate 100 uniformly distributed discrete numbers on the set 1:n |

To model rolling of a fair die, the sample space \( \mathcal{S}=\left\{1,2,3,4,5,6\right\}\). A roll of the die should generate a number from the sample space with equal probability. This single event can be modelled in matlab as

1 |
>> r = ceil(6.*rand(1,1)); |

(The uniformity test for discrete random numbers can be done using the same code given above in the previous section)

#### Note:There is one more function in Matlab called **“randn”**. This function is used to generate random numbers that are normally distributed.

## Reference:

## See Also:

[1] Introduction to Random Variables, PDF and CDF

[2] Chi-Squared distribution

[3] Derivation of Error Rate Performance of an optimum BPSK receiver in AWGN channel

[4] Eb/N0 Vs BER for BPSK over Rician Fading Channel

[5] BER Vs Eb/N0 for QPSK modulation over AWGN

[6] BER Vs Eb/N0 for 8-PSK modulation over AWGN

[7] Simulation of M-PSK modulation techniques in AWGN channel

[8] Performance comparison of Digital Modulation techniques

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