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Classic Graphing 84 manual · Chapter 16 of 25

Probability distributions

Normal, t, chi-square, F, binomial, Poisson and geometric probabilities, inverse normal and t, and shading areas on a graph.

16.1The DISTR menu#

Press 2ndDISTRvars. The DISTR tab lists the distributions; the DRAW tab shades areas. Choosing a function opens a small form, where you fill the fields (use ▼ to move) and finish on Paste, which writes the finished command on the home screen. Press enter again to get the answer. Entries are remembered for next time.

The DISTR menu. The display shows: [DISTR] DRAW / 1:normalpdf( / 2:normalcdf( / 3:invNorm( / 4:invT( / 5:tpdf( / 6:tcdf( / 7↓χ²pdf(.
The DISTR menu.
The normalcdf form: lower, upper, mean and standard deviation. The display shows: normalcdf / lower: / upper: / μ:0 / σ:1 / Paste.
The normalcdf form: lower, upper, mean and standard deviation.
ItemCommandGives
1normalpdf(Height of the normal curve at x
2normalcdf(Probability between lower and upper (normal)
3invNorm(The x for a given left-tail area; Tail LEFT, CENTER or RIGHT
4invT(The t for a given left-tail area
5, 6tpdf(, tcdf(Student t curve height and probability
7, 8χ²pdf(, χ²cdf(Chi-square curve height and probability
9, 0Fpdf(, Fcdf(F curve height and probability
A, Bbinompdf(, binomcdf(Exactly x successes, and x or fewer
C, Dpoissonpdf(, poissoncdf(Exactly x events, and x or fewer
E, Fgeometpdf(, geometcdf(First success on trial x, and by trial x

16.2Find a normal probability#

P(a≤X≤b)=∫ab1σ2πe−(x−μ)22σ2 dxP(a\le X\le b)=\int_a^b \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\,dx
Example 16.1

Within one standard deviation

P(−1≤Z≤1)≈0.6827P(-1\le Z\le1)\approx0.6827

  1. Open DISTR item 2, type −1 for lower, down, 1 for upper, then down three times to Paste and press enter.

    2ndDISTRvars2(−)1▼1▼▼▼enter

  2. Press enter to calculate the pasted command.

    enter
Result of the example "Within one standard deviation" The display shows: normalcdf(⁻1,1,0,1) / .6826894921.
You should seenormalcdf(⁻1,1,0,1) .6826894921
Example 16.2

Below 130 when the mean is 100 and sd 15

P(X≤130), μ=100, σ=15P(X\le130),\ \mu=100,\ \sigma=15

  1. Use lower −1ᴇ99 (a huge negative number), upper 130, μ 100, σ 15.

    2ndDISTRvars2(−)12ndEE,99▼130▼100▼15▼enterenter
Result of the example "Below 130 when the mean is 100 and sd 15" The display shows: normalcdf(⁻1ᴇ99,130,100,15 / ) / .9772498681.
You should see.9772498681
Example 16.3

Height of the standard normal curve at 0

  1. DISTR item 1, type 0 for x, go down to Paste and enter twice.

    2ndDISTRvars10▼▼▼enterenter
Result of the example "Height of the standard normal curve at 0" The display shows: normalpdf(0,0,1) / .3989422804.
You should see.3989422804

16.3Find a value from a probability (inverse normal and t)#

Example 16.4

The 97.5th percentile

P(Z≤z)=0.975⇒z≈1.96P(Z\le z)=0.975\Rightarrow z\approx1.96

  1. DISTR item 3: type .975 for area, move down four times to Paste, enter twice.

    2ndDISTRvars3.975▼▼▼▼enterenter
Result of the example "The 97.5th percentile" The display shows: invNorm(.975,0,1) / 1.959963985.
You should see1.959963985
Example 16.5

Both tails (CENTER)

  1. Type .95, go down three times to Tail, choose CENTER with right and enter, then down to Paste and enter twice.

    2ndDISTRvars3.95▼▼▼▶enter▼enterenter
Result of the example "Both tails (CENTER)" The display shows: invNorm(.95,0,1,CENTER) / {⁻1.959963985 1.959963985}.
You should see{⁻1.959963985 1.959963985}
Example 16.6

Inverse t

  1. DISTR item 4: area .975 and 10 degrees of freedom.

    2ndDISTRvars4.975▼10▼enterenter
Result of the example "Inverse t" The display shows: invT(.975,10) / 2.228138852.
You should see2.228138852

16.4Binomial, Poisson, t and chi-square probabilities#

P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k}
The binomial probability for k successes in n trials.
Example 16.7

Five heads in ten tosses

(105)(12)10\binom{10}{5}\left(\tfrac12\right)^{10}

  1. Press 2nd vars then alpha math (A: binompdf). Type 10 trials, .5 for p, 5 for x, then Paste and enter.

    2ndDISTRvarsalphaAmath10▼.5▼5▼enterenter
Result of the example "Five heads in ten tosses" The display shows: binompdf(10,.5,5) / .24609375.
You should see.24609375
Example 16.8

Five or fewer heads

  1. binomcdf( is alpha apps (B).

    2ndDISTRvarsalphaBapps10▼.5▼5▼enterenter
Result of the example "Five or fewer heads" The display shows: binomcdf(10,.5,5) / .623046875.
You should see.623046875
Example 16.9

Poisson probability

P(X=0)=e−2P(X=0)=e^{-2}

  1. poissonpdf( is alpha prgm (C): mean 2, x 0.

    2ndDISTRvarsalphaCprgm2▼0▼enterenter
Result of the example "Poisson probability" The display shows: poissonpdf(2,0) / .1353352832.
You should see.1353352832
Example 16.10

Geometric, first success by trial 3

  1. geometcdf( is alpha cos (F): p .5, x 3.

    2ndDISTRvarsalphaFcos.5▼3▼enterenter
Result of the example "Geometric, first success by trial 3" The display shows: geometcdf(.5,3) / .875.
You should see.875
Example 16.11

t probability

  1. tcdf( is item 6: lower −2, upper 2, df 10.

    2ndDISTRvars6(−)2▼2▼10▼enterenter
Result of the example "t probability" The display shows: tcdf(⁻2,2,10) / .9266119652.
You should see.9266119652
Example 16.12

Chi-square probability

  1. χ²cdf( is item 8: lower 0, upper 3.84, df 1.

    2ndDISTRvars80▼3.84▼1▼enterenter
Result of the example "Chi-square probability" The display shows: χ²cdf(0,3.84,1) / .9499564788.
You should see.9499564788

16.5Shade an area under a curve#

The DRAW tab (press 2ndDISTRvars then ▶) has ShadeNorm(, ShadeT(, Shadeχ²( and ShadeF(. Type the lower and upper limits and the distribution values, and the area is drawn and measured.

Example 16.13

Shade the middle 68%

  1. Press 2nd vars, right, 1 (ShadeNorm(), type −1,1), enter.

    2ndDISTRvars▶1(−)1,1)enter
Result of the example "Shade the middle 68%" The display shows: Area=.6826894921 / low=⁻1 up=1.
You should seeArea=.6826894921 low=⁻1 up=1

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