Bernoulli trials and the Bernoulli distribution

An experiment produces A, B or C. Define X=1 for A and X=0 for B or C. Is X a Bernoulli variable?

Bernoulli trials and moments

A Bernoulli variable is a specified binary indicator, recording 1 on its success event and 0 otherwise. Distinguish that variable from raw outcome labels and from a sequence of trials

1. 1Name the recorded variable A specified 0/1 indicator is Bernoulli. Its raw experiment may have several outcomes that the coding combines \(\displaystyle X=1\text{ on }A,\quad X=0\text{ on }A^{\prime}\)
2. 2Define the event coding The event coded 1 determines the success parameter. Outcome probabilities alone do not choose the event label \(\displaystyle p=\Pr(A)\)
3. 3Use the complement The failure probability is one minus the success probability \(\displaystyle p=\frac{3}{5},\quad 1-p=\frac{2}{5}\)
4. 4Use the moment rules For a Bernoulli variable, the mean is p, the variance is p(1-p), and the standard deviation is the principal square root \(\displaystyle \operatorname{E}(X)=p,\quad \operatorname{Var}(X)=p(1-p),\quad \operatorname{SD}(X)=\sqrt{p(1-p)}\)

L and R each have probability 1/2. One will be coded 1 and the other 0, but the choice is not supplied. What is known?

X records 0, 1 or 2, all with positive probability. I records 1 when X=1 and 0 otherwise. Which statement is correct?

Independent binary indicators X₁ and X₂ have success probabilities 1/3 and 2/3. Do they form a common-probability binomial trial model?

Defined Bernoulli variable

\(\displaystyle X=1\text{ on success},\quad X=0\text{ otherwise},\quad\Pr(X=1)=\frac35\)

Find the failure probability from the success probability

Defined Bernoulli variable

\(\displaystyle X=1\text{ on success},\quad X=0\text{ otherwise},\quad\Pr(X=1)=\frac35\)

Use p and q to find the variance

For X=1 on success and X=0 otherwise, with success probability 3/5, which mean calculation is correct?

Derive the mean from the two recorded values

The success value 1 receives weight p and the failure value 0 receives weight 1−p. The expected value is therefore p, not its complement

Use the fact that squaring 0 and 1 leaves them unchanged

For a binary indicator X²=X, so E(X²)=p. Subtract the square of the mean to obtain the variance. Standard deviation is the nonnegative square root

Variance and root condition

\(\displaystyle p(1-p)=\frac6{25},\qquad p<\frac12\)

Find p for the Bernoulli variable with variance 6/25, given p<1/2.

Keep the root condition with the equation

The probabilities p and 1−p give the same variance. A condition selects a root when the roots differ. At p=1/2 the roots coincide and the variance is 1/4. At p=0 or p=1 the distribution is constant and has zero variance

Define and calculate a complete indicator model, ungraded

Use the stated outcome probabilities and indicator coding. Write its full distribution, calculate its mean, variance and exact standard deviation, and explain why three raw outcomes do not prevent I from being Bernoulli. Then explain what changes if a different variable records A, B and C as 0, 1 and 2.

Check the recorded support before using Bernoulli moments

Raw outcomes A, B and C have probabilities 1/6, 1/3 and 1/2. I records 1 for A or B and 0 for C; a different variable records A, B and C as 0, 1 and 2. For I, the success probability is the sum of the probabilities for A and B, giving equal probabilities at the two recorded values. Its mean is 1/2, variance 1/4 and standard deviation 1/2. The different raw-value variable has three recorded values, so its moments must be calculated from its own three-value distribution, not the Bernoulli formula

Separate the coding, distribution and repetition questions

A specified binary indicator is Bernoulli. Raw categories may be grouped by an explicit coding, but a separately recorded three-valued variable stays different. Missing event labels need not imply an unknown parameter when the two probabilities are equal. Independent repetition requires a common parameter for the binomial trial model