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CIVL 181 Tutorial 3

CIVL 181 Tutorial 3. PMF to CDF PDF to CDF Mean and s.d., c.o.v. Formulae for reference. P (T B = t). 0.6. 0.2. 0.2. t. 4. 5. 6. P (T A = t). activity B. 0.5. 0.3. 0.2. t. 3. 4. 5. activity A. From PMF to CDF. 3.1 Two constructions A, B have the PMF below:

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CIVL 181 Tutorial 3

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  1. CIVL 181 Tutorial 3 • PMF to CDF • PDF to CDF • Mean and s.d., c.o.v. • Formulae for reference

  2. P (TB = t) 0.6 0.2 0.2 t 4 5 6 P (TA = t) activity B 0.5 0.3 0.2 t 3 4 5 activity A From PMF to CDF 3.1 Two constructions A, B have the PMF below: Assume activity A and B are s.i., draw PMF and CDF of the total no. of days required.

  3. From PMF to CDF 1. Observe the possible values (7 ≤ X ≤ 11) 2. Compute the associated probabilities of each value by theorem of total probability

  4. From PMF to CDF P (X7) = P(A3) P(B4) = 0.3 x 0.2 = 0.06 P (X8) = P(A3)P(B5) + P(A4)P(B4) P (X9) = P(A3)P(B6) + P(A4)P(B5) + P(A5)P(B4)…… P(X7) = 0.06; P(X8) = 0.28; P(X9) = 0.4; P(X10) = 0.22; P(X11) = 0.04; P(X7) + P(X8) + P(X9) + P(X10) + P(X11) = ?

  5. P (X) 0.4 0.28 0.22 0.06 0.04 t 7 8 9 10 11 PMF • CDF as a practice

  6. From PDF to CDF Suppose a PDF follows a square relationship with its variable x, x ranges from 3 to 6. Draw its PDF and CDF.

  7. From PDF to CDF • f(x) = x2, 3 ≤ x ≤ 6? • The first thing to do is CHECK! • Should setup f(x) as f(x) = kx2 and find k.

  8. f(x) 36/63 9/63 x 3 6 k? • Area property: the area of PDF is 1

  9. CDF • Wrong! • Because F(1) = 1 + 9/63 • There are integration constants • Use boundary condition: • C = - 9 / 63

  10. F(x) 1 x 3 6 Remember! • CDF always starts from 0 and end at 1 • Integration const may exist, especially when x are not start from 0. • Use boundary condition to eliminate

  11. Mean and s.d., c.o.v. • Q1: If daily temperature is 20oc, is that suitable for human to leave? The place may be a desert with 40oc in daytime and 00c at night • Q2: For two r.v. X and Y, if X has 1000 times larger s.d. than Y, is that X more dispersed? Think of expressing concerte strength in terms of MPa and kPa (and compare the numerical value)

  12. Formulae for reference • Discrete: • Continuous: • s.d. = • Median: • Skewness, Kurtosis very seldom used.

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