Tutorial 2

1. Tutorial 2 Probability Density Functions & Normal Density

2. Probability Density Function • Unlike discrete random variables, continuous r.v. can take infinite no. of values. • In this case, the probability that the r.v. takes a particular value is zero, i.e. P(X = x) = 0 • We can’t represent the distribution of X by probability mass function like the discrete case.

3. Probability Density Function • We use probability density function f(x) to represent the distribution of a continuous r.v. • The value of f(x) is not a probability. Instead, the integral of f(x) gives the required probability. • Several important properties can identify a p.d.f.

4. Properties of p.d.f. • f(x) 0

5. Ex. of p.d.f. • Ex. (P.71 Q.2) Suppose you choose a real no. X from the interval [2,10] with a density function of the form f(x) = Cx, where C is a constant. (a) Find C.

6. Ex. of p.d.f. (b) Find P(E), where E=[a,b] is a subinterval of [2,10].

7. Ex. of p.d.f. (c) Find P(X > 5), P(X < 7) and P(X2 -12X + 35 > 0) .

8. Normal Density • One commonly used distribution is the normal density. Its p.d.f is given by where - < X <  . • mean =  , std. dev. = 

9. Normal Density • Assume X is a normal r.v. with mean =  and std. dev. = . • Let Z = (X - )/ . • Z is normal r.v. with mean = 0 & std. dev. = 1 • We said Z has a standard normal distribution. • As it is hard to obtain the prob. of a normal distribution from the integral of its p.d.f., P(Z<a) has been calculated in a table for use.

10. Normal Density • By transforming any normal r.v. X to Z, P(X < b) can be obtained from the table, i.e. P(X < b) = P((X - )/ < (b - )/) = P(Z < (b - )/) = P(Z < a)

11. Normal Density • Ex.(P.222 Q.25(d)) Let X be a r.v. normally distributed with  = 70,  = 10. Estimate P(60 < X < 80).