Constructing a Binomial Probability Mass Function

00:00 6 marks

Consider a binomial random variable \(X\) with parameters \(n=5\) and \(p=0.4\). \(X \sim B(5, 0.4)\)

(a) Construct the probability mass function (pmf) table by calculating \(P(X=k)\) for \(k=0,1,2,3,4,5\).

(b) Verify that the sum of all probabilities in the table is equal to 1.

Part a

5 minutes 3 marks

Question

Given a binomial random variable \(X\) with \(n=5\) and \(p=0.4\), construct the probability mass function (pmf) table by computing \(P(X=k)=\binom{5}{k}(0.4)^k(0.6)^{5-k}\) for each \(k=0,1,2,3,4,5\). Provide the values rounded to five decimal places.

Hint Hint

Hint

Recall that \(\binom{n}{k}=\frac{n!}{k!(n-k)!}\). Compute each term step by step and organize your results in a table format.

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Complete Answer

P(X=0)=0.07776, P(X=1)=0.25920, P(X=2)=0.34560, P(X=3)=0.23040, P(X=4)=0.07680, P(X=5)=0.01024

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Part b

3 minutes 3 marks

Question

Using the pmf table from part (a), verify that the sum of the probabilities \(P(X=0)+P(X=1)+\cdots+P(X=5)\) is equal to 1. Show your calculation.

Hint Hint

Hint

Add all the computed probabilities from the table and explicitly show the sum to confirm it equals 1.

Answer Answer
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Answer Answer

Complete Answer

0.07776 + 0.25920 + 0.34560 + 0.23040 + 0.07680 + 0.01024 = 1.00000

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