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Find the Likelihood Function for a Discrete Random Variable with Given pmf and Observations
Statistics
University Year 1
Question Content
Let X be a discrete random variable with the pmf as given below, for 0 < θ < 2. \n\n| X | 3 | 4 | 5 | 6 |\n|----|----|----|----|----|\n| P(X=x) | θ/3 | (2-θ)/3 | θ/6 | (2-θ)/6 |\n\nSix independent observations were taken from this distribution, as follows: 4, 3, 3, 5, 5, 6.\n\n(a) Obtain the likelihood function, L(θ), and show the steps of your work.
Correct Answer
L(θ) = (θ^4 (2-θ)^2) / 648
Detailed Solution Steps
1
Step 1: Identify the frequency of each observed value. From the observations 4, 3, 3, 5, 5, 6: the value 3 occurs 2 times, 4 occurs 1 time, 5 occurs 2 times, and 6 occurs 1 time.
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Step 2: Recall that for independent observations, the likelihood function is the product of the probability mass function (pmf) evaluated at each observation. This can be simplified to the product of each pmf raised to the power of its observed frequency.
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Step 3: Substitute the pmf values and their corresponding frequencies: L(θ) = [P(X=3)]^2 × [P(X=4)]^1 × [P(X=5)]^2 × [P(X=6)]^1
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Step 4: Plug in the pmf expressions from the table: L(θ) = (θ/3)^2 × ((2-θ)/3)^1 × (θ/6)^2 × ((2-θ)/6)^1
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Step 5: Expand and combine like terms: First calculate the coefficients: (1/3^2 × 1/3 × 1/6^2 × 1/6) = 1/(9×3×36×6) = 1/648. Then combine the θ terms: θ^2 × θ^2 = θ^4, and the (2-θ) terms: (2-θ)^1 × (2-θ)^1 = (2-θ)^2.
6
Step 6: Combine the results to get the final likelihood function: L(θ) = (θ^4 (2-θ)^2) / 648, for 0 < θ < 2.
Knowledge Points Involved
1
Probability Mass Function (pmf)
A pmf is a function that gives the probability that a discrete random variable is exactly equal to some value. For a discrete random variable X, P(X=x) is non-negative for all x, and the sum of all P(X=x) over all possible x equals 1. It is used to model discrete probability distributions and is the foundation for calculating likelihood functions for discrete data.
2
Likelihood Function
For a set of independent observations, the likelihood function L(θ) is the product of the pmf (or pdf for continuous variables) evaluated at each observation, where θ is the parameter of the distribution being estimated. It represents the plausibility of different values of θ given the observed data, and is used in maximum likelihood estimation to find the value of θ that best fits the data.
3
Independent Observations in Statistics
Observations are independent if the outcome of one observation does not affect the outcome of any other. For independent observations, the joint probability of the entire set of observations is the product of the individual probabilities of each observation, which is a key property used to construct likelihood functions.
4
Algebraic Simplification of Products
When simplifying the product of terms with exponents, we use the rule a^m × a^n = a^(m+n) for like bases. This is essential for condensing the product of pmf terms into a simplified likelihood function, making it easier to analyze and use for parameter estimation.
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