Let [latex]X=[/latex] the number of computer components tested until the first defect is found.

[latex]X[/latex] takes on the values 1, 2, 3, … where [latex]p = 0.02[/latex].

[latex]X{\sim}G(0.02)[/latex]

Find [latex]P(x=7)[/latex]. [latex]P(x=7)=0.0177[/latex].

USING THE TI-83, 83+, 84, 84+ CALCULATOR

To find the probability that [latex]x=7[/latex],

• Enter 2nd, DISTR
• Scroll down and select geometpdf(
• Press ENTER
• Enter 0.02, 7); press ENTER to see the result: [latex]P(x=7)=0.0177[/latex]

To find the probability that [latex]x\leq7[/latex], follow the same instructions EXCEPT select E:geometcdf(as the distribution function.

The probability that the seventh component is the first defect is 0.0177.

The graph of [latex]X{\sim}G(0.02)[/latex] is:

The [latex]y[/latex]-axis contains the probability of [latex]x[/latex], where [latex]X=[/latex] the number of computer components tested.

The number of components that you would expect to test until you find the first defective one is the mean, [latex]\displaystyle{\mu}={50}[/latex].

The formula for the mean is [latex]\displaystyle{\mu}=\frac{{1}}{{p}}=\frac{{1}}{{0.02}}={50}[/latex]

The formula for the variance is [latex]\displaystyle{\sigma}^{{2}}={(\frac{{1}}{{p}})}{(\frac{{1}}{{p}}-{1})}={(\frac{{1}}{{0.02}})}{(\frac{{1}}{{0.02}}-{1})}={2},{450}[/latex]

The standard deviation is [latex]\displaystyle{\sigma}=\sqrt{{{(\frac{{1}}{{p}})}{(\frac{{1}}{{p}}-{1})}}}=\sqrt{{{(\frac{{1}}{{0.02}})}{(\frac{{1}}{{0.02}}-{1})}}}={49.5}[/latex]

1. [latex]P(x=10)=\text{geometpdf}(0.0128,10)=0.0114[/latex]
2. [latex]P(x=20)=\text{geometpdf}(0.0128,20)=0.01[/latex]
   1. [latex]\text{Mean}={\mu}=\frac{{1}}{{p}}=\frac{{1}}{{0.0128}}={78}[/latex]
   2. [latex]\text{Standard Deviation}={\sigma}=\sqrt{{\frac{{{1}-{p}}}{{{p}^{{2}}}}}}=\sqrt{{\frac{{{1}-{0.0128}}}{{0.0128}^{{2}}}}}\approx{77.6234}[/latex]