FS1-3 Geometric and negative binomial distributions — 9FM0 spec coverage

FS1-3.1

Geometric and negative binomial distributions.

For independent Bernoulli trials with success probability $p$ , a geometric variable $X$ counts the trial of the first success: $P(X=x)=p(1-p)^{x-1}$ for $x=1,2,\ldots$ .
A negative binomial variable $Y$ counting the trial of the $r$ th success has $P(Y=y)=\binom{y-1}{r-1}p^r(1-p)^{y-r}$ for $y=r,r+1,\ldots$ .
The combination $\binom{y-1}{r-1}$ places the first $r-1$ successes before the final, fixed success on trial $y$ .
A common error is to use $\binom{y}{r}$ , forgetting that the last trial must be a success.

Tier 1 · Easy

2 marks

ORIGINAL

Independent trials have success probability $0.3$ . Find the probability that the first success occurs on trial $4$ .

Tier 2 · Standard

3 marks

ORIGINAL

Independent trials have success probability $0.4$ . Find the probability that the third success occurs on trial $5$ .

Tier 3 · Hard

4 marks

ORIGINAL

Each route attempt succeeds independently with probability $0.35$ . Calculate the probability that the second successful route is completed by the fifth attempt.

FS1-3.2

Mean and variance of a geometric distribution with parameter p.

For a geometric variable that counts the trial of the first success, $E(X)=1/p$ and $\operatorname{Var}(X)=(1-p)/p^2$ .
The standard deviation is $\sqrt{1-p}/p$ , obtained by taking the positive square root of the variance.
For example, when $p=0.25$ , the mean is $4$ trials and the variance is $12$ .
A common error is to use the alternative convention that counts failures before the first success; the Edexcel FS₁ convention starts at $X=1$ .

Tier 1 · Easy

2 marks

ORIGINAL

The random variable $X$ is geometric with parameter $p=0.2$ and counts the trial of the first success. Find its mean and variance.

Tier 2 · Standard

4 marks

ORIGINAL

A geometric random variable has mean $8$ . Find $p$ , its variance and its standard deviation.

Tier 3 · Hard

6 marks

ORIGINAL

For a geometric random variable $X$ , the variance is twice the mean. Determine $p$ , then find $P(X\geq4)$ , $E(X)$ and $\operatorname{Var}(X)$ .

FS1-3.3

Mean and variance of negative binomial distribution.

If $X$ counts the trial of the $r$ th success, then $E(X)=r/p$ and $\operatorname{Var}(X)=r(1-p)/p^2$ .
The ratio $\operatorname{Var}(X)/E(X)=(1-p)/p$ is useful when the mean and variance are given and $p$ is unknown.
For example, $r=3$ and $p=0.6$ give mean $5$ and variance $10/3$ .
A common error is to use the formulas for the number of failures, whose mean is $r(1-p)/p$ ; FS₁ defines $X$ as the trial number of the $r$ th success.

Tier 1 · Easy

2 marks

ORIGINAL

The random variable $X$ counts the trial on which the fourth success occurs, with success probability $0.25$ . Find $E(X)$ and $\operatorname{Var}(X)$ .

Tier 2 · Standard

6 marks

ORIGINAL

A negative binomial random variable has mean $15$ and variance $30$ . Find $p$ and $r$ , then calculate $P(X=7)$ .

Tier 3 · Hard

5 marks

ORIGINAL

Successive trials are independent with success probability $0.4$ . Let $X$ be the trial on which the sixth success occurs. Find the probability that $X\leq8$ .