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Probability Distributions

A probability distribution assigns probabilities to possible outcomes. The expected value is the long-run weighted average.

$$E(X)=\sum xp(x), \qquad Var(X)=E(X^2)-[E(X)]^2$$

Expected Value and Variance

Let $X=0,1,2$ with probabilities $0.20,0.50,0.30$. Find $E(X)$ and $Var(X)$.

$$E(X)=0(0.20)+1(0.50)+2(0.30)=1.10$$
$$E(X^2)=0^2(0.20)+1^2(0.50)+2^2(0.30)=1.70$$
$$Var(X)=1.70-(1.10)^2=0.49$$

Final answer: $E(X)=1.10$ and $Var(X)=0.49$.

Problem: Binomial Exact Count

A concrete cylinder has an 80% chance of passing a strength check. If 5 cylinders are tested, find the probability that exactly 4 pass.

$$P(X=4)=\binom{5}{4}(0.80)^4(0.20)=0.4096$$

Answer: The probability is 0.4096.

Problem: Poisson Arrival Count

An inspection desk receives an average of 3 requests per hour. Find the probability of exactly 2 requests in one hour.

$$P(X=2)=e^{-3}\frac{3^2}{2!}=0.2240$$

Answer: The probability is about 0.224.

Problem: Normal Distribution z-Score

Survey errors are normally distributed with mean 0 mm and standard deviation 5 mm. Find the z-score for an error of 8 mm.

$$z=\frac{x-\mu}{\sigma}=\frac{8-0}{5}=1.60$$

Answer: The z-score is 1.60.

Problem: Expected Value of a Discrete Variable

A repair cost is P1,000 with probability 0.30 and P4,000 with probability 0.70. Find the expected repair cost.

$$E(X)=1000(0.30)+4000(0.70)=3100$$

Answer: The expected repair cost is P3,100.

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