Geometric Random Variables

Solomon Xie

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Statistical Guess

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Jan 13, 2019

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Refer to wiki: Geometric distribution
Refer to Khan academy: Geometric random variables introduction

Geometric vs. Binomial Random Variables

• A binomial setting has a set number of trials, and the variable in question is the number of successes that occur in those trials.
• A geometric setting DOES NOT have a set number of trials, and the variable in question is the number of trials it takes to get the first success.

In both settings, the trials are independent and the probability of success remains the same on each trial.

The only difference between Geometric R.V. and Binomial R.V. is that,
The Geometric R.V. DOES NOT have a certain number of trails.

Requirements of Geometric R.V.:

• p: Same probability p on each trail.
• Yes-no question: Each trail's outcome is either success or failure.
• Independent: Each trail is independent to each other.

Geometric Probability

The geometric distribution gives the probability that the first occurrence of success requires k independent trials, each with success probability p.

If it’s asking for Number of Trails, then the number Trails = failures + success = (n-1) + 1.
If it's asking for Number of failures, then the number Trails = Failures+1 = n+1.
And that's why the formula is slightly different.

Assume p is the probability of success on each trail, n is the number of trails or failures:

• Number of Trials until the first success:

• Number of Failures until the first success:

Example

Solve:

Mean & Variance of Geometric Probability

Cumulative Geometric Probability

We know how to calculate Geometric Probability at each value, but Cumulative G.P.would be bit tricky.

The formula literally means: FAIL a TIMES IN A ROW.

This formula is good for X>a, and with a bit twist you can get most out of it.
etc.,

• P(X≥4) is the same with P(X>3)
• P(X≤5) is the same with 1 - P(X>5)
• P(X<7) is the same with 1 - P(X>6)

Example

Solve:

• The easiest way is to apply the cumulative geometric probability formula:
• P(X<5) = 1 - P(X>4) = 1 - Failure⁴ = 1-0.9^4 = 0.34
• Another way is to calculate each item in the sequence:

P(C≤4) = P(C=1) + P(C=2) + P(C=3) + P(C=4)

• which we expand it to:

P(C≤4) = 0.9⁰*0.1 + 0.9*0.1 + 0.9²*0.1 + 0.9³*0.1

• We could easily recognize it as a standard Geometric Series, so we can apply the formula:

P(C≤4) = 0.1 * (1-0.9⁴) / (1-0.9) = 0.34

Example

Solve:

• It’s the same as “the probability of 5 failures in a row.”

Example

Solve:

-