Cal11 calculator
Search calculator...

Calculate E X 3 Where X Has The Following Pmf

Reviewed by Calculator Editorial Team

This calculator helps you compute the expected value of e^(x³) when x is a random variable with a given probability mass function (PMF). The calculation involves summing the probabilities multiplied by the transformed values of x.

What is a Probability Mass Function (PMF)?

A probability mass function (PMF) describes the probability distribution of a discrete random variable. For a discrete variable x that can take values x₁, x₂, ..., xₙ, the PMF p(x) gives the probability that x equals each possible value:

p(x) = P(X = x) for all x in the sample space

The PMF must satisfy two conditions:

  1. 0 ≤ p(x) ≤ 1 for all x
  2. The sum of all probabilities must equal 1: Σ p(x) = 1

Common discrete distributions include the binomial, Poisson, and geometric distributions, each with their own PMF formulas.

Calculation Method

To calculate the expected value of e^(x³) given a PMF, we use the definition of expected value for a discrete random variable:

E[e^(x³)] = Σ [p(x) * e^(x³)] for all x in the sample space

This formula involves:

  1. Identifying all possible values of x and their corresponding probabilities p(x)
  2. Calculating e^(x³) for each x
  3. Multiplying each transformed value by its probability
  4. Summing all these products to get the expected value

Note: The expected value E[e^(x³)] is not the same as e^[E(x³)]. The exponential function is not linear, so these two expressions yield different results.

Worked Example

Let's calculate E[e^(x³)] for a random variable x with the following PMF:

x p(x)
-1 0.2
0 0.5
1 0.3

Step-by-step calculation:

  1. Calculate e^(x³) for each x:
    • e^(-1³) = e⁻¹ ≈ 0.3679
    • e^(0³) = e⁰ = 1
    • e^(1³) = e¹ ≈ 2.7183
  2. Multiply each transformed value by its probability:
    • 0.2 * 0.3679 ≈ 0.0736
    • 0.5 * 1 = 0.5
    • 0.3 * 2.7183 ≈ 0.8155
  3. Sum the products: 0.0736 + 0.5 + 0.8155 ≈ 1.3891

The expected value E[e^(x³)] is approximately 1.3891 for this distribution.

Interpreting the Result

The calculated expected value represents the average value of e^(x³) over many trials of the experiment. Here's what this means:

  • If you repeat the experiment many times, the average of e^(x³) will approach this expected value
  • It provides a central tendency measure for the transformed random variable
  • The result depends on both the original distribution of x and the transformation e^(x³)

Important: The expected value is not a prediction for any single trial. It's a long-term average across many independent trials.

Frequently Asked Questions

What's the difference between PMF and PDF?

A probability mass function (PMF) is used for discrete random variables, while a probability density function (PDF) is used for continuous variables. The PMF gives exact probabilities for specific values, while the PDF gives relative likelihoods over ranges.

Can I use this calculator for continuous variables?

No, this calculator is specifically for discrete variables with a probability mass function. For continuous variables, you would need to use the expected value formula for a probability density function.

What if my PMF doesn't sum to 1?

The calculator will normalize your probabilities by dividing each value by the sum of all probabilities. This ensures the PMF meets the mathematical requirements for a valid probability distribution.