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Chapter 1: Problem 1

Let \(X\) equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of \(X\) and compute the probability that \(X\) is equal to an odd number.

Short Answer

Expert verified
The probability mass function, pmf of \(X\) is calculated by plugging \(n=4\) and \(p = 0.5\) into the binomial distribution formula for each outcome. The probability that \(X\) is an odd number is the sum of the probabilities \(P(X = 1)\) and \(P(X = 3)\).

Step by step solution

01

Determine the Binomial Distribution

We begin by recognizing that this is a binomial distribution problem with parameters n = 4 (number of trials) and p = 0.5 (probability of getting 'head' in each trial). The probability mass function (pmf) of a binomial distribution is given by the formula: \(P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}\) where k is the number of 'heads', 'n' is the number of trials, and 'p' is the probability of success on each trial.
02

Calculate pmf for Each Outcome

Using the pmf formula from step 1, we calculate the pmf for all possible values of \(X\), which are 0, 1, 2, 3, and 4. For instance, to calculate for \(k=1\), we substitute \(n=4\), \(k=1\), and \(p=0.5\) into the formula to obtain \(P(X = 1) = \binom{4}{1}0.5^1(1-0.5)^{4-1}\). We repeat this process for \(k = 0, 2, 3, 4\).
03

Compute Probability for X Being an Odd Number

Finally, to calculate the probability of \(X=1\) or \(X=3\) (i.e. \(X\) being an odd number), we simply add the probabilities we already calculated for these outcomes: \(P(X = \text{{odd}}) = P(X = 1) + P(X = 3)\).

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Most popular questions from this chapter

Let \(X\) have a Cauchy distribution which has the pdf $$ f(x)=\frac{1}{\pi} \frac{1}{x^{2}+1}, \quad-\infty

Cast a die a number of independent times until a six appears on the up side of the die. (a) Find the pmf \(p(x)\) of \(X\), the number of casts needed to obtain that first six. (b) Show that \(\sum_{x=1}^{\infty} p(x)=1\). (c) Determine \(P(X=1,3,5,7, \ldots)\). (d) Find the cdf \(F(x)=P(X \leq x)\).

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