# Random fields & Probability cheat sheet

• Random variable – the value of a random experiment.
• Random field – a spatial function that assigns a random variable to each position
• Distribution function of a random variable (aka Cumulative distribution function, CDF)
– Consider $$P_f(z)=\mathcal{P} \{ f <z\}$$ The equation tells us how likely for the random variable $$f$$ to be less than $$z$$. $$P_f(z)$$ is the distribution function of $$f$$. $$\mathcal{P}$$ is the probability.
• Probability of an event happening has the properties:
• It is a non-negative number
• The probability of all the events (all possible outcomes) must equal one
• The probability of a random variable taking a specific real value is zero. Loosely, this is because a real values need an infinite precision to describe them. We can however specify the probability over an infinity small range. Combining those ranges over all possible values give the probability density function.
• The probability density function is the differential of the CDF
$$p_f(z)=\frac{ d P_f(z) }{dz}$$
• The expected or mean value is $$\mu_f=E\{f\}=\int_{-\infty}^{\infty} zp_f(z) dz$$
• The variance is $$\sigma^2_f=E\{(f-\mu_f)^2\}=\int_{-\infty}^{\infty} (z-\mu_f)^2p_f(z) dz$$
• If we have many random variables we can define a joint distribution function
$$P_{f_1,f_2,f_3,\ldots,f_n}(z_1,z_2,z_3 \ldots z_n)=\mathcal{P} \{ f_1 <z_1,f_2 <z_2,f_3 <z_3,\ldots,f_n <z_n \}$$
• and the joint probability density function
$$p_{f_1,f_2,f_3,\ldots,f_n}(z_1,z_2,z_3 \ldots z_n)=\frac{P_{f_1,f_2,f_3,\ldots,f_n}(z_1,z_2,z_3 \ldots z_n)}{dz_1 dz_2 dz_3 \ldots dz_n}$$
• If random variables are independent then
$$P_{f_1,f_2,f_3,\ldots,f_n}(z_1,z_2,z_3 \ldots z_n)=P_{f1}(z_1)P_{f2}(z_2)P_{f3}(z_3)\ldots P_{fn}(z_n)$$
• If they are uncorrelated then
$$E\{f_i f_j\}=E\{f_i \}E\{f_j\} \quad , \forall i,j, i \neq j$$
• and orthogonal if
$$E\{f_i f_j\}=0$$
• The covariance is $$c_{ij}=E\{(f_i-\mu_{f_i})( f_j-\mu_{f_j})\}$$ which equals zero if they are uncorrelated.

Since we are interested in images (in this blog) we shall introduce random fields from this point of view.  If we have a random variable at every point $$\mathbf{r}$$ in a 2D space we have a random field. The random field can be written $$f(\mathbf{r},\omega_i)$$ where $$\omega_i$$ is the outcome. For a fixed $$\omega_i$$ the random field gives a grey level image as we scan over all $$\mathbf{r}$$. Scanning over all possible $$\omega_i$$ gives a series of images.