Unit 6: Probability and Stochastic Processes

ECE183 — Mathematics For Robotics 6 min read

I. Orientation

Probability provides a mathematical language for uncertainty, incomplete information, sensor noise, and random disturbances. In robotics, a robot rarely knows its exact position, velocity, or environment state; instead, it represents possible values and their likelihoods. A stochastic process extends this idea to quantities that change randomly over time, such as noisy measurements or motion disturbances.

Random experiment: An operation with an uncertain outcome, such as measuring distance with a laser sensor.
Sample space: The set of all possible outcomes, denoted by (\Omega).
Event: A subset of the sample space, denoted by (A), (B), or similar symbols.
Probability: A numerical measure (P(A)) satisfying (0 \leq P(A) \leq 1), (P(\Omega)=1), and additivity for mutually exclusive events.
Conditional probability: The probability of (A) given (B), written (P(A\mid B)), and defined only when (P(B)>0).
Independence: Events (A) and (B) are independent when (P(A\cap B)=P(A)P(B)).
Random variable: A function mapping outcomes to numerical values, such as a robot’s measured range (X).
Stochastic-process viewpoint: A random variable indexed by time, written (X_t), represents changing uncertain quantities.

II. Bayes' theorem — Updating beliefs with evidence

Bayes' theorem gives the probability of a hypothesis after observing evidence. It is fundamental to robot localization, diagnosis, mapping, and sensor fusion because it combines prior knowledge with new measurements.

A. Bayes' theorem

Bayes' theorem states that, for events (A) and (B) with (P(B)>0),

Formal statement: The posterior probability of (A) after observing (B) is
TEXT
```
  P(A | B) = [P(B | A) P(A)] / P(B)
```
Here, (P(A\mid B)) is the posterior, (P(B\mid A)) is the likelihood, (P(A)) is the prior, and (P(B)) is the evidence.
Evidence expansion: If (A_1,\ldots,A_n) are mutually exclusive and exhaustive hypotheses,
TEXT
```
  P(B) = Σ_i P(B | A_i) P(A_i)
```
The sum accounts for every possible cause of the observation.
Robot interpretation: A map-based position estimate is the prior; a sensor reading is evidence; the updated position distribution is the posterior.
Worked example: Suppose a robot is either in region (A) with prior probability (0.6) or outside it with probability (0.4). A sensor detects a landmark with probability (0.9) in (A) and (0.2) outside (A). Then
TEXT
```
  P(A | landmark) = (0.9 × 0.6) / [(0.9 × 0.6) + (0.2 × 0.4)]
                  = 0.54 / 0.62
                  ≈ 0.871
```
The landmark raises belief in region (A) from (0.6) to approximately (0.871).

III. Probability distributions and probability mass functions — Discrete uncertainty

A probability distribution describes how probability is assigned to the possible values of a random variable. For a discrete variable, this assignment is represented by a probability mass function.

A. Probability distributions and probability mass functions

This topic distinguishes the general distribution of a random variable from the discrete function that gives the probability of each exact value.

Probability distribution: For a random variable (X), the distribution specifies probabilities for all possible values of (X).
Probability mass function (PMF): For a discrete (X), the PMF is
TEXT
```
  p_X(x) = P(X = x)
```
The value (p_X(x)) is the probability that (X) equals (x).
Normalization: A valid PMF satisfies
TEXT
```
  Σ_x p_X(x) = 1
```
where the sum covers every allowed value of (x).
Event probability: For a set (S) of discrete values,
TEXT
```
  P(X ∈ S) = Σ_{x ∈ S} p_X(x)
```
Non-negativity: Every PMF value obeys (p_X(x)\geq 0).
Robotics example: If (X) is the number of faulty sensor readings in five measurements, then (X) can take only (0,1,2,3,4,5); its PMF assigns a probability to each count.
Discrete versus continuous: A discrete variable has individual point probabilities, whereas a continuous variable generally has (P(X=x)=0) for every exact value.

IV. Cumulative distribution functions — Probability up to a threshold

The cumulative distribution function records the probability that a random variable is less than or equal to a specified value. It works for both discrete and continuous variables.

A. Cumulative distribution functions

The CDF is useful when a robot needs threshold probabilities, such as the chance that position error is below a tolerance.

Definition: For any random variable (X), its CDF is
TEXT
```
  F_X(x) = P(X ≤ x)
```
Here, (x) is a possible numerical threshold.
Discrete form: If (X) is discrete,
TEXT
```
  F_X(x) = Σ_{t ≤ x} p_X(t)
```
The CDF accumulates all PMF values at or below (x).
Interval probability: For (a<b),
TEXT
```
  P(a < X ≤ b) = F_X(b) - F_X(a)
```
Properties: A CDF is non-decreasing, bounded between (0) and (1), approaches (0) as (x\to-\infty), and approaches (1) as (x\to\infty).
Worked example: If (P(X=0)=0.2), (P(X=1)=0.5), and (P(X=2)=0.3), then (F_X(1)=P(X\leq1)=0.2+0.5=0.7).
Robotic use: (F_X(0.1)=0.95) can mean that a localization error is no more than (0.1) m with probability (95\%).

V. Probability density functions — Continuous uncertainty

A probability density function describes the relative concentration of probability for a continuous random variable. It is not itself the probability of an exact value.

A. Probability density functions

PDFs model quantities such as wheel-slip error, range noise, and robot position, which can vary over continuous intervals.

Definition: A continuous random variable (X) has PDF (f_X(x)) satisfying
TEXT
```
  P(a ≤ X ≤ b) = ∫_a^b f_X(x) dx
```
Here, (a) and (b) are interval boundaries.
Normalization: The total area under a PDF equals one:
TEXT
```
  ∫_{-∞}^{∞} f_X(x) dx = 1
```
Non-negativity: A PDF must satisfy (f_X(x)\geq0).
Exact values: For a continuous variable, (P(X=x)=0); probability is assigned to intervals, not isolated points.
CDF relationship: Where the derivative exists,
TEXT
```
  f_X(x) = dF_X(x)/dx
```
Conversely, (FX(x)=\int{-\infty}^{x}f_X(t)\,dt).
Units: If (X) is measured in metres, (f_X(x)) has units of inverse metres, while the integrated probability is dimensionless.
Robotics use: A sharply concentrated PDF represents a precise sensor; a broad PDF represents high measurement uncertainty.

VI. Mean and variance — Numerical summaries of uncertainty

The mean identifies the central or expected value of a random variable, while variance measures the spread around that value.

A. Mean and variance

These quantities summarize distributions and are widely used in estimation, filtering, and motion modeling.

Discrete mean: For a discrete (X),
TEXT
```
  μ = E[X] = Σ_x x p_X(x)
```
Here, (E[X]) is expectation and (\mu) is the mean.

Continuous mean: For a continuous (X),

TEXT

  μ = E[X] = ∫_{-∞}^{∞} x f_X(x) dx

Variance: Variance is
TEXT
```
  Var(X) = σ² = E[(X - μ)²]
```
Here, (\sigma^2) measures average squared deviation from (\mu).
Computational form: The equivalent identity is
TEXT
```
  σ² = E[X²] - μ²
```
Standard deviation: (\sigma=\sqrt{\sigma^2}) has the same units as (X), making it easier to interpret than variance.
Worked example: If (X) is (0) with probability (0.5) and (2) with probability (0.5), then (\mu=1), (E[X^2]=2), and (\sigma^2=2-1^2=1).
Interpretation: A mean position of (3) m with standard deviation (0.2) m indicates concentration near (3) m; the mean alone does not describe that uncertainty.

VII. Binomial distribution — Fixed trials with two outcomes

The binomial distribution models the number of successes in a fixed number of independent trials when each trial has the same success probability.

A. Binomial distribution

It is appropriate for repeated yes/no events, such as successful detections across a known number of sensor scans.

Assumptions: There are (n) fixed trials; each trial has success probability (p); failure probability is (1-p); and trials are independent.
PMF: If (X) counts successes,
TEXT
```
  P(X = k) = C(n,k) p^k (1-p)^(n-k)
```
Here, (k\in{0,\ldots,n}), and (C(n,k)=n!/[k!(n-k)!]).
Mean and variance:
TEXT
```
  E[X] = np
  Var(X) = np(1-p)
```
Worked example: For (n=4) scans and detection probability (p=0.75), the probability of exactly (k=3) detections is
TEXT
```
  P(X=3) = C(4,3)(0.75)^3(0.25) = 0.421875
```
Limitation: Independent, identical trials are essential; changing sensor conditions or correlated detections may invalidate the model.

VIII. Poisson distribution — Counts over an interval

The Poisson distribution models the number of events occurring in a fixed interval when events occur independently at a constant average rate.

A. Poisson distribution

It is useful for counting arrivals, failures, obstacles, or communication events over time or distance.

Assumptions: Events occur independently, the average rate is constant, and simultaneous multiple events are negligibly likely in an infinitesimally small interval.
PMF: For event count (X) and rate parameter (\lambda>0),
TEXT
```
  P(X = k) = e^(-λ) λ^k / k!
```
Here, (k=0,1,2,\ldots), and (\lambda) is the expected count in the chosen interval.
Mean and variance:
TEXT
```
  E[X] = λ
  Var(X) = λ
```
Equality of mean and variance is a defining feature of the Poisson model.
Rate scaling: If events occur at rate (r) per second during time (T), then (\lambda=rT). For (r=2) events/s and (T=3) s, (\lambda=6).
Relation to binomial: A Poisson distribution approximates a binomial distribution when (n) is large, (p) is small, and (np\approx\lambda).
Limitation: Bursty, dependent, or time-varying event arrivals require a different model.

IX. Normal distribution — Continuous bell-shaped uncertainty

The normal distribution is a continuous, symmetric model characterized by its mean and variance. It often approximates accumulated independent measurement errors.

A. Normal distribution

Normal models are central in robotics because many sensor and estimation errors are treated as approximately Gaussian.

PDF: A normal random variable (X\sim\mathcal N(\mu,\sigma^2)) has
TEXT
```
  f_X(x) = [1 / (σ√(2π))] exp[-(x-μ)²/(2σ²)]
```
Here, (\mu) is the mean, (\sigma>0) is the standard deviation, and (\sigma^2) is the variance.
Shape: The curve is symmetric about (x=\mu); its peak occurs at (\mu), and larger (\sigma) produces greater spread.
Standardization: The standardized variable is
TEXT
```
  Z = (X - μ) / σ
```
It follows the standard normal distribution (\mathcal N(0,1)).
Empirical rule: Approximately (68\%) of values lie within (\mu\pm\sigma), (95\%) within (\mu\pm2\sigma), and (99.7\%) within (\mu\pm3\sigma).
Worked interpretation: If range error is (\mathcal N(0,0.01^2)) m, then about (95\%) of errors lie between (-0.02) m and (0.02) m.
Robotics limitation: Real errors may be skewed, bounded, or contain outliers; a normal approximation should be checked against sensor behavior rather than assumed automatically.