# randn

Normally distributed random numbers

## Description

X = randn returns a random scalar drawn from the standard normal distribution.

example

X = randn(n) returns an n-by-n matrix of normally distributed random numbers.

example

X = randn(sz1,...,szN) returns an sz1-by-...-by-szN array of random numbers where sz1,...,szN indicate the size of each dimension. For example, randn(3,4) returns a 3-by-4 matrix.

example

X = randn(sz) returns an array of random numbers where size vector sz defines size(X). For example, randn([3 4]) returns a 3-by-4 matrix.

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X = randn(___,typename) returns an array of random numbers of data type typename. The typename input can be either "single" or "double". You can use any of the input arguments in the previous syntaxes.

example

X = randn(___,"like",p) returns an array of random numbers like p; that is, of the same data type and complexity (real or complex) as p. You can specify either typename or "like", but not both.

X = randn(s,___) generates numbers from random number stream s instead of the default global stream. To create a stream, use RandStream. You can specify s followed by any of the input argument combinations in previous syntaxes.

## Examples

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Generate a 5-by-5 matrix of normally distributed random numbers.

r = randn(5)
r = 5×5

0.5377   -1.3077   -1.3499   -0.2050    0.6715
1.8339   -0.4336    3.0349   -0.1241   -1.2075
-2.2588    0.3426    0.7254    1.4897    0.7172
0.8622    3.5784   -0.0631    1.4090    1.6302
0.3188    2.7694    0.7147    1.4172    0.4889

Generate values from a bivariate normal distribution with specified mean vector and covariance matrix.

mu = [1 2];
sigma = [1 0.5; 0.5 2];
R = chol(sigma);
z = repmat(mu,10,1) + randn(10,2)*R
z = 10×2

1.5377    0.4831
2.8339    6.9318
-1.2588    1.8302
1.8622    2.3477
1.3188    3.1049
-0.3077    1.0750
0.5664    1.6190
1.3426    4.1420
4.5784    5.6532
3.7694    5.2595

Save the current state of the random number generator and create a 1-by-5 vector of random numbers.

s = rng;
r = randn(1,5)
r = 1×5

0.5377    1.8339   -2.2588    0.8622    0.3188

Restore the state of the random number generator to s, and then create a new 1-by-5 vector of random numbers. The values are the same as before.

rng(s);
r1 = randn(1,5)
r1 = 1×5

0.5377    1.8339   -2.2588    0.8622    0.3188

Create a 3-by-2-by-3 array of random numbers.

X = randn([3,2,3])
X =
X(:,:,1) =

0.5377    0.8622
1.8339    0.3188
-2.2588   -1.3077

X(:,:,2) =

-0.4336    2.7694
0.3426   -1.3499
3.5784    3.0349

X(:,:,3) =

0.7254   -0.2050
-0.0631   -0.1241
0.7147    1.4897

Create a 1-by-4 vector of random numbers whose elements are single precision.

r = randn(1,4,"single")
r = 1x4 single row vector

0.5377    1.8339   -2.2588    0.8622

class(r)
ans =
'single'

Create a matrix of normally distributed random numbers with the same size as an existing array.

A = [3 2; -2 1];
sz = size(A);
X = randn(sz)
X = 2×2

0.5377   -2.2588
1.8339    0.8622

It is a common pattern to combine the previous two lines of code into a single line.

X = randn(size(A));

Create a 2-by-2 matrix of single-precision random numbers.

p = single([3 2; -2 1]);

Create an array of random numbers that is the same size and data type as p.

X = randn(size(p),"like",p)
X = 2x2 single matrix

0.5377   -2.2588
1.8339    0.8622

class(X)
ans =
'single'

Generate 10 random complex numbers from the standard complex normal distribution.

a = randn(10,1,"like",1i)
a = 10×1 complex

0.3802 + 1.2968i
-1.5972 + 0.6096i
0.2254 - 0.9247i
-0.3066 + 0.2423i
2.5303 + 1.9583i
-0.9545 + 2.1460i
0.5129 - 0.0446i
0.5054 - 0.1449i
-0.0878 + 1.0534i
0.9963 + 1.0021i

By default, randn(n,"like",1i) generates random numbers from the standard complex normal distribution. The real and imaginary parts are independent normally distributed random variables with mean 0 and variance 1/2. The covariance matrix is of the form [1/2 0; 0 1/2].

z = randn(50000,1,"like",1i);
cov_z = cov(real(z),imag(z),1)
cov_z = 2×2

0.4980    0.0007
0.0007    0.4957

To specify a more general complex normal distribution, define the mean and covariance matrix. For instance, specify the mean as $\mu =1+2\mathrm{i}$ and the covariance matrix as $\sigma =\left[\begin{array}{cc}{\sigma }_{\mathrm{xx}}& {\sigma }_{\mathrm{xy}}\\ {\sigma }_{\mathrm{yx}}& {\sigma }_{\mathrm{yy}}\end{array}\right]=\left[\begin{array}{cc}2& -2\\ -2& 4\end{array}\right]$.

mu = 1 + 2i;
sigma = [2 -2; -2 4];

Transform the previously generated data to follow the newly defined complex normal distribution. Include the factor of sqrt(2) when scaling the data because the variance for the real and imaginary parts in the original distribution is 1/2.

R = chol(sigma);
z_comp = [real(z) imag(z)];
z = repmat(mu,50000,1) + z_comp*sqrt(2)*R*[1; 1i];
z(1:10)
ans = 10×1 complex

1.7604 + 3.8331i
-2.1945 + 6.4138i
1.4508 - 0.3002i
0.3868 + 3.0977i
6.0606 + 0.8560i
-0.9090 + 8.2011i
2.0259 + 0.8850i
2.0108 + 0.6993i
0.8244 + 4.2823i
2.9927 + 2.0115i

## Input Arguments

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Size of square matrix, specified as an integer value.

• If n is 0, then X is an empty matrix.

• If n is negative, then it is treated as 0.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Size of each dimension, specified as separate arguments of integer values.

• If the size of any dimension is 0, then X is an empty array.

• If the size of any dimension is negative, then it is treated as 0.

• Beyond the second dimension, randn ignores trailing dimensions with a size of 1. For example, randn(3,1,1,1) produces a 3-by-1 vector of random numbers.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Size of each dimension, specified as a row vector of integer values. Each element of this vector indicates the size of the corresponding dimension:

• If the size of any dimension is 0, then X is an empty array.

• If the size of any dimension is negative, then it is treated as 0.

• Beyond the second dimension, randn ignores trailing dimensions with a size of 1. For example, randn([3 1 1 1]) produces a 3-by-1 vector of random numbers.

Example: sz = [2 3 4] creates a 2-by-3-by-4 array.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Data type (class) to create, specified as "double", "single", or the name of another class that provides randn support.

Example: randn(5,"single")

Prototype of array to create, specified as a numeric array.

Example: randn(5,"like",p)

Data Types: single | double
Complex Number Support: Yes

Random number stream, specified as a RandStream object.

Example: s = RandStream("dsfmt19937"); randn(s,[3 1])

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### Standard Real and Standard Complex Normal Distributions

When generating random real numbers, the randn function generates data that follows the standard normal distribution:

$f\left(x\right)=\frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}.$

Here, x is a random real variable with mean 0 and variance 1.

When generating random complex numbers, such as when using the command randn(...,"like",1i), the randn function generates data that follows the standard complex normal distribution:

$f\left(z\right)=\frac{1}{\pi }{e}^{-{|z|}^{2}}.$

Here, z is a random complex variable whose real and imaginary parts are independent normally distributed random variables with mean 0 and variance 1/2.

## Tips

• The sequence of numbers produced by randn is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn. You can control that shared random number generator using rng.

## Version History

Introduced before R2006a

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