Unit 6: Probability and Stochastic Processes - Subjective Questions

Bayes' theorem relates a conditional probability to its reverse conditional probability:

For a robot localization problem, let represent a possible robot location and represent a sensor measurement. Then:

• is the prior probability of the robot being at location .
• is the likelihood, or probability of obtaining measurement when the robot is at .
• is the total probability of observing the measurement.
• is the posterior probability, representing the updated belief about the robot's location.

Thus, Bayes' theorem combines prior knowledge with sensor evidence to produce an improved estimate of the robot's state.

Let denote the event that an obstacle is present and denote the event that the sensor reports an obstacle.

Given:

Using the law of total probability:

Applying Bayes' theorem:

Therefore, the probability that an obstacle is actually present is approximately , or .

A random variable is a function that assigns a numerical value to each outcome of a random experiment.

A discrete random variable can take a finite or countably infinite number of distinct values. Its probabilities are represented by a probability mass function (PMF).

• Example: The number of obstacles detected by a robot in one scan can be .
• Example: The number of successful wireless transmissions in a fixed number of attempts.

A continuous random variable can take any value within an interval. Its probabilities are represented using a probability density function (PDF).

• Example: The distance measured by a range sensor.
• Example: The angular error in a robot's orientation estimate.

For a discrete variable, may be positive for individual values. For a continuous variable, the probability of one exact value is generally zero, so probabilities are calculated over intervals.

A probability mass function (PMF) gives the probability that a discrete random variable takes a particular value :

A valid PMF must satisfy:

• Non-negativity:
  
• Total probability equals one:
  
• Probability of an event: For a set of possible values,
  

For example, suppose a robot's proximity sensor reports the number of nearby obstacles as , with:

All probabilities are non-negative and:

Therefore, these values define a valid PMF.

The cumulative distribution function (CDF) of a random variable is defined as:

It gives the probability that takes a value less than or equal to .

Important properties of a CDF are:

• .
• is non-decreasing.
• The CDF is right-continuous.

For a discrete random variable, the CDF is a step function. If has PMF , then:

For a continuous random variable with PDF :

The CDF is useful in robotics for calculating probabilities such as the chance that sensor error remains below a specified tolerance.

For a continuous random variable , the probability density function (PDF) is denoted by and the cumulative distribution function (CDF) is denoted by .

The CDF is obtained by integrating the PDF from to :

If the CDF is differentiable, differentiating both sides gives:

The probability that lies between and is:

Equivalently:

The PDF itself is not a probability at a single point. Instead, the area under the PDF over an interval represents probability. A valid PDF must satisfy and:

The mean, or expected value, describes the average or central value of a random variable.

For a discrete random variable:

For a continuous random variable:

The variance measures the spread of values around the mean:

An equivalent formula is:

The standard deviation is:

For robot sensor measurements:

• The mean represents the expected measured distance or angle.
• The variance indicates the uncertainty or noise level of the sensor.
• A small variance means measurements are closely concentrated around the mean.
• A large variance means the sensor readings are more uncertain.

Therefore, mean and variance are important for probabilistic state estimation and sensor fusion.

Let , where and are constants.

The mean of is:

Using the linearity of expectation:

Therefore, if :

For the variance:

Substituting and :

Hence:

Thus:

• Adding shifts the mean but does not change the variance.
• Multiplying by scales the mean by and the variance by .

These rules are useful when converting sensor units or transforming robot coordinates.

The binomial distribution models the number of successes in a fixed number of independent trials, where every trial has only two possible outcomes: success or failure.

If is the number of successes in trials and the probability of success in each trial is , then:

Its PMF is:

The assumptions are:

• The number of trials is fixed.
• Each trial has two outcomes.
• The probability of success is constant.
• The trials are independent.

In robotics, a binomial model can represent the number of successful sensor detections in a fixed number of scans or the number of successful communication attempts among repeated transmissions.

Let be the number of correct identifications. Then:

The binomial PMF is:

For exactly 6 correct identifications:

Since:

we obtain:

Therefore, the probability of exactly 6 correct identifications is approximately , or .

Represent the binomial random variable as the sum of indicator variables:

where for success and for failure. Each indicator has:

Using linearity of expectation:

For a Bernoulli indicator:

Because the trials are independent, the variances add:

Therefore:

Thus, for a binomial random variable:

• Mean:
• Variance:
• Standard deviation:

The Poisson distribution models the number of times an event occurs in a fixed interval of time, distance, area, or volume when events occur independently at a constant average rate.

If is the number of events and the average rate is , then:

Its PMF is:

The Poisson model is appropriate when:

• Events occur independently.
• The average event rate is constant.
• Two events are unlikely to occur at exactly the same instant.
• The probability of an event in a very small interval is proportional to the interval size.

In robotics, it may model the number of obstacles encountered per meter, the number of sensor failures per hour, or the number of communication requests received per second.

Let be the number of obstacles detected in the next 100 meters. The average rate is:

Therefore:

The Poisson PMF is:

For exactly 2 obstacles:

Therefore, the probability of detecting exactly 2 obstacles is approximately , or .

For , the PMF is:

The expected value is:

The term for is zero. Using :

Since the sum equals :

Using the factorial moment:

Therefore:

The variance is:

Thus, the mean and variance of a Poisson distribution are both equal to .

The binomial and Poisson distributions are both discrete probability models, but they describe different types of counting processes.

In robotics:

• The binomial distribution can model the number of successful detections in a fixed number of sensor readings.
• The Poisson distribution can model the number of obstacles encountered during a fixed travel distance.

The Poisson distribution can approximate a binomial distribution when is large, is small, and remains moderate.

The normal distribution is a continuous probability distribution with a symmetric, bell-shaped curve centered at its mean.

A random variable follows a normal distribution with mean and variance if:

Its PDF is:

The parameters have the following meanings:

• determines the center or location of the distribution.
• determines the spread or variance.
• is the standard deviation.

The total area under the curve is one, and the mean, median, and mode are all equal to . Normal distributions are commonly used to model measurement noise, localization errors, and deviations in robot motion.

If:

then the standardized variable is defined as:

The resulting variable follows the standard normal distribution:

Standardization converts a value into the number of standard deviations it lies above or below the mean. The standard normal CDF, denoted by , gives:

For a value :

Similarly:

A standard normal table or calculator is used to obtain the required values of .

Let the sensor error be . We are given:

We need to find:

Standardize the endpoints:

Therefore:

Using the standard normal CDF:

Using symmetry, and :

Thus, the probability that the absolute error is less than cm is approximately , or .

The empirical rule, also called the 68-95-99.7 rule, states that for a normally distributed random variable:

• Approximately of observations lie within one standard deviation of the mean:
  
• Approximately lie within two standard deviations:
  
• Approximately lie within three standard deviations:
  

For robot sensor errors, if the error is approximately normal, this rule helps estimate the expected proportion of measurements within a tolerance range. For example, a measurement error outside is highly unusual and may indicate an outlier, sensor malfunction, or unexpected environmental condition.

The central limit theorem (CLT) states that, under suitable conditions, the sum or average of a large number of independent random variables tends to have an approximately normal distribution, even when the individual variables are not normally distributed.

If are independent and identically distributed with mean and variance , then for sufficiently large :

is approximately normally distributed as:

The standardized form is:

In robotics, the CLT is useful because:

• Repeated noisy sensor readings can be averaged.
• The distribution of accumulated motion or measurement errors can often be approximated by a normal distribution.
• It supports confidence intervals and statistical decision-making.
• It provides a theoretical basis for Gaussian-based estimation methods.