New Zealand
Level 8 - NCEA Level 3

# Bernoulli mean and variance

Lesson

An earlier chapter defined the Bernoulli distribution.

A Bernoulli trial is essentially a single trial of a binomial experiment. Its properties provide an easy way of deducing the properties of a binomial distribution because of a theorem in mathematical statistics that says:

if a random variable $Y$Y is the sum of random variables $X_1,X_2,X_3,...$X1,X2,X3,... , then the mean of $Y$Y is the sum of the means of the $X_i$Xis, and if the $X_i$Xis are independent, then their variances add to give the variance of $Y$Y .

So, given that the mean value of a Bernoulli random variable is $p$p, we deduce that the mean value of a binomial random variable defined as the number of successes in a sequence of $n$n Bernoulli trials, must be $np$np.

Similarly, given that the variance of a Bernoulli random variable is $p\left(1-p\right)$p(1p), and a binomial experiment consists of $n$n of these Bernoulli trials, which are independent, the corresponding binomial random variable must have variance $np\left(1-p\right)$np(1p).

#### Example

Consider an experiment in which a trial is repeated $20$20 times. The probability of success in this experiment is $0.4$0.4. We define a random variable $Y$Y to be the number of successes in this sequence of $20$20 Bernoulli trials.

Here, we would say that $Y$Y is a binomial random variable and the parameters of its distribution are $20$20 and $0.4$0.4. This is summarised by the notation $Y\sim B\left(20,0.4\right)$Y~B(20,0.4)

We can understand the random variable $Y$Y as the sum $X_1+X_2+...+X_{20}$X1+X2+...+X20 of twenty Bernoulli random variables $X_i$Xi, each of which can be thought of as a binomial random variable with a single trial. Thus, $X_i\sim B\left(1,0.4\right)$Xi~B(1,0.4).

Each of the Bernoulli random variables $X$X must have mean $0.4$0.4 and variance $p\left(1-p\right)=0.4\times0.6=0.24$p(1p)=0.4×0.6=0.24.

#### Worked Examples

##### QUESTION 1

Ten Bernoulli trials are conducted and the expected value is $9$9.

1. State the probability of success.

2. Calculate the variance of these ten trials.

##### QUESTION 2

In a certain electorate, $12%$12% of voters usually vote for the L.E.F.T party.

In the upcoming election, there are $6000$6000 people enrolled to vote.

1. How many are expected to vote for the L.E.F.T party?

2. Calculate the standard deviation for the number of voters who do not vote for the L.E.F.T party.

##### QUESTION 3

James has a cabbage patch that has been attacked by cabbage moth. He expects to have $5$5 of his cabbages ruined with a standard deviation of $2$2.

1. Solve the number of cabbages, $n$n, James planted and the probability, $p$p, of each being struck by the cabbage moth.

Write both solutions on the same line, separated by a comma.

2. Given that more than $2$2 were ruined, what is the probability that at most $4$4 were ruined?

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91586

Apply probability distributions in solving problems