This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.

Cumulative normal distribution (z table)

Exercises

• A binary message $m$, where $m$ is equal either to $0$ or to $1$, is sent over an information channel. Because of noise in the channel, the message received is $X$, where $X = m + E$, and $E$ is a random variable representing the channel noise. Assume that if $X ≤ 0.5$ then the receiver concludes that $m = 0$ and that if $X > 0.5$ then the receiver concludes that $m = 1$. Assume that $E ∼ N(0, 0.25)$.

  a. If the true message is m = 0, what is the probability of an error, that is, what is the probability that the receiver concludes that $m = 1$?

  b. Let $\sigma^2$ denote the variance of E. What must be the value of $\sigma^2$ so that the probability of error when $m = 0$ is $0.01$?

Solution:

• Chebyshev’s inequality states that for any random variable $X$ with mean $\mu$ and variance $ \sigma^2 $,

  and for any positive number $k$, $ P(\mid X − \mu \mid ≥ k \sigma) ≤ 1∕k^2$. Let $X ∼ N(\mu, \sigma^2)$.

  (a) Compute $P(\mid X − 𝜇 \mid ≥ k \sigma)$ for the values $k = 1, 2, ~ and ~ 3$.

  (b) Are the actual probabilities close to the Chebyshev bound of $1∕k^2$, or are they much smaller?

Solution:

• In a certain university, math SAT scores for the entering freshman class averaged 650 and had a standard deviation of 100. The maximum possible score is 800.

  Is it possible that the scores of these freshmen are normally distributed? Explain.