Probability review (week 2)

1 Bernoulli, Binomial, Poisson and normal distributions. In this excercise we deal with Bernoulli, binomial, Poisson and normal random variables (RVs). A Bernoulli RV X models experiments, such as a coin toss, where success happens with probability p and failure with probability 1 − p. Success is indicated by X = 1 and failure by X = 0. Therefore, the probability mass function (pmf) of X is
P {X = 0} = 1 − p, P {X = 1} = p

(1)

A binomial random variable (RV) with parameters (n, p) counts the number of successes in n independent Bernoulli trials that succeed with probability p. Thus, we can write a Binomial RV Y as n Y = i=1 Xi

(2)

where the Xi are Bernoulli RVs with pmfs as in (1). The pmf of a binomial RV is easily derived by noting that we have X = x for some integer x between 0 and 1 if and only there are x successful Bernoulli trials – something that happens with probability px – and n − x failed experiments – which happens with probability (1 − p)n−x – and that there are n different ways in which this could happen. x Thus n x n! px (1 − p)n−x , x = 0, 1, . . . , n. (3) p(x) := P {X = x} = p (1 − p)n−x = (n − x)!x! x A Poisson RV X takes values in the nonnegative integers. We say that X is Poisson with parameter λ it its pmf is λx p(x) = e−λ , x = 0, 1, . . . (4) x! Different from the other two, a normal random variable X can take any real value (not just 0 or 1 like the Bernoulli or integers between 0 and n for the binomial or nonnegative integers for the Poisson). We say X is a continuous RV. Probabilities are described now using a probability distribution function (pdf). For a normal RV with mean µ and variance σ 2 , the pdf is 1 2 2 f (x) = √ e−(x−µ) /2σ , x ∈ R (5) 2πσ Another concept of interest here is that of a cumulative distribution function (cdf) defined as the probability of X not exceeding x, i.e., F