EX 3.33 purchase with credit card

• Scenario:

  • In a certain store, 75% of all purchases are made with a credit card.
  • Let $X$ represent the number of purchases made with a credit card among 10 randomly selected purchases.

• Random variable distribution:

  • $X$ follows a binomial distribution:
  $$X\sim \mathrm{Bin}(10,0.75)$$

• Expected value:

  • The expected value (mean) of $X$:
  $$E(X)=np=10\times 0.75=7.5$$

• Variance:

  • The variance of $X$:
  $$V(X)=npq=10\times 0.75\times 0.25=1.875$$

• Standard deviation:

  • The standard deviation of $X$:
  $$\sigma =\sqrt{1.875}\approx 1.37$$

• Interpretation:

  • Although $X$ can only take on integer values (0 to 10), the expected value $E(X)$ need not be an integer.
  • When performing a large number of independent binomial experiments (each with $n=10$ trials and $p=0.75$), the average number of purchases made with a credit card per experiment will be close to 7.5.

The probability that $X$ is within 1 standard deviation of its mean value is

$$\begin{array}{rl}& P(7.5-1.37\le X\le 7.5+1.37)\\ & =P(6.13\le X\le 8.87)\\ & =P(X=7\text{ or }8)\\ & =0.532\end{array}$$