How to find the mean of a probability distribution

What is the difference between expected value and arithmetic mean?

The expected value derives from the full probability model and accounts for unequal probabilities, while the arithmetic (sample) mean treats each observed value equally without probabilities. The sample mean serves as an estimate of E(X) through the law of large numbers as the sample size grows. Expected value is used for predictions under known distributions, and sample mean is used for descriptive statistics or inference from data.

How do you calculate the mean of a discrete probability distribution?

To calculate the mean of a discrete probability distribution, multiply each possible value by its corresponding probability and sum all the products. The formula is:

$\mu =$

where x_i represents each possible value of the random variable X and $P(X=x_i)$ represents the corresponding probability.

Step-by-step procedure for discrete distributions

1. List all possible discrete values \(x_i\) of the random variable and their probabilities $P(X=x_i)$ in a table, ensuring the probabilities sum to 1.
2. Compute each product $x_i·P(X=x_i)$ for every value.
3. Sum all the products to obtain the mean μ.

Discrete distribution calculation example

For a random variable X with P(X = 2) = 0.16, P(X = 3) = 0.45, P(X = 4) = 0.32, and P(X = 5) = 0.07:

$\mu =$

How do you calculate the mean of a continuous probability distribution?

To calculate the mean of a continuous probability distribution, integrate the product of each value and its probability density over the entire support of the distribution. The formula is:

$\mu =E($

where f(x) is the probability density function (PDF).

Step-by-step procedure for continuous distributions

1. Identify the PDF f(x) and its domain. Verify that the integral of f(x) over its domain equals 1.
2. Form the integrand x · f(x).
3. Compute the definite integral over the support of the distribution.
4. Evaluate using integration techniques such as integration by parts. Software like R or Python assists with complex cases.

Why integration replaces summation

Integration replaces discrete summation because continuous variables take uncountably infinite values. The integral acts as a continuous analog to the discrete sum, weighting each value x by the infinitesimal density f(x)dx to yield the long-run average.

What are the mean formulas for common probability distributions?

Standard probability distributions have simplified formulas for calculating their means directly from distribution parameters.

Discrete distribution means

Binomial distribution:

$\mu =np$

where n is the number of trials and p is the probability of success on each trial.

Poisson distribution:

$\mu =\lambda$

where λ is the average rate of events. The mean equals the variance in a Poisson distribution.

Continuous distribution means

Normal distribution:

$\mu =\mu$

The mean is the location parameter μ, which specifies the center of symmetry.

Exponential distribution:

$\mu =\frac{1}{\lambda }$

where λ is the rate parameter. This represents the mean time between events.

Uniform distribution on [a, b]:

$\mu =\frac{a+b}{2}$

he mean is the midpoint of the interval.

What are the key properties of expected value?

The expected value has several properties that simplify calculations without requiring the full distribution.

Linearity of expectation

For constants a and b and any random variables X and Y (independent or dependent):

$E(aX+bY)=aE(X)+bE(Y)$

his property allows decomposition of complex expressions into simpler components.

Constants and scaling

The expected value of a constant equals the constant:

$E(c)=c$

Multiplying a random variable by a constant scales the expected value by that constant:

$E(cX)=cE(X)$

Sums and products

The expected value of a sum equals the sum of expected values:

$E(X+Y)=E(X)+E(Y)$

For independent random variables X and Y, the expected value of a product equals the product of expected values:

$E(XY)=E(X)\cdot E(Y)$