Question

10. Explain in words, and words only, the following properties of a cumulative density function (CDF) F(x). Explain the meaning of any numerical values. (a) F(X) = P(X <r) (b) P (X<) = P(XSI) (c) F(---) = 0 (d) F(0) = 1 (e) if ro <, then F(r) > F(ta) (f) P (X > 1) = 1 - F(X) (9) P ( SX S16) -F ) - F ( )

Answer 1

the f) Bow Now, for a particular value 'se, X either assumes values greater than or equal to &ror X assumes values strictly less than P (x>x) + P ( x 4 x ) =1. → P(X>x) = 1- P(Xex) [using (b)] al-F(M) [ Using (a) ]

Here X is a random variable a) CDF of x = F(x) = Priobability that x assumes values less than or equal to 'x' b) Probability that x assumes values strictly Probability that x takes less than 'x' values less than or equal to x c) Probability that x takes values less than or equal to a lie. X can't take any value less than or equal to .o. d) Probability that x takes values less than or equal to to. always takes l.e. X values less than to requality is not possible ahore F(x) = P(X<x) e) if a value da is less than a I then Probability that x takes values less than or equal to all is greater than prob. that x takes values less than or equal to B Fina) holds] ola FM).

values in between and including Ha&to and g) Prob. that & takes values in between is equal to Fb) - F(Xa). - Feb k PCX EXER) Flora)

Here X is a random variable a) CDF of x = F(x) = Priobability that x assumes values less than or equal to 'x' b) Probability that x assumes values strictly Probability that x takes less than 'x' values less than or equal to x c) Probability that x takes values less than or equal to a lie. X can't take any value less than or equal to .o. d) Probability that x takes values less than or equal to to. always takes l.e. X values less than to requality is not possible ahore F(x) = P(X<x) e) if a value da is less than a I then Probability that x takes values less than or equal to all is greater than prob. that x takes values less than or equal to B Fina) holds] ola FM).

values in between and including Ha&to and g) Prob. that & takes values in between is equal to Fb) - F(Xa). - Feb k PCX EXER) Flora)