Begin2.DVI

If each event isequally likely to happen, then one usually assigns a probability valueP(ei) =^1 nto each event as then ∑n i=1P ...

of the probabilities associated with the simple events common to both the eventsE 1 andE 2 are counted twice. The sum of these c ...

IfE 2 is the event that an 11 is rolled, then P(E 2 ) =P((5,6)) +P((6,5)) = 2/36 = 1/ 18 IfE 3 is the event doubles are rolled, ...

Conditional Probability If two events E 1 and E 2 are related in some manner such that the probability of occurrence of event E ...

Example 11-3. Given an ordinary deck of 52 cards, suppose it is required to find the probability of selecting two cards and they ...

Example 11-5. Two cards are selected at random from an ordinary deck of 52 cards. Find the probability that both cards are spade ...

Example 11-6. How many three digit even numbers can be formed using the digits { 1 , 2 , 3 , 5 , 7 }if repetition of any digits ...

In general, the number of permutations of nthings taken mat a time is given by the formula nPm=n(n−1)(n−2) ···(n−m+ 1) (11 .32) ...

Binomial Coefficients The binomial expansion (p+q)n, for nan integer, can be expressed in the form (p+q)n= ( n 0 ) pn+ ( n 1 ) p ...

In general, in studying n-trials associate with a two event happening, one would examine the binomial expansion (p+q)n= ∑n j=0 ( ...

The function F(x) = ∑ xj≤x f(xj)is called the cumulative frequency function asso- ciated with the discrete sample. In the contin ...

Table 11.4 Mean and Variance for Discrete and Continuous Distributions Discrete Continuous population μ=E[x] = ∑n j=1 xjf(xj) me ...

and these quantities are called the kth central moments of X. Note the special cases E[1] = 1, μ =E[X], σ^2 =E[(X−μ)^2 ] (11 .45 ...

and the variance on the Z-scale is given by σ∗^2 = ∫∞ −∞ (z−μ∗)^2 f∗(z)dz = ∫∞ −∞ z^2 f∗(z)dz since μ∗= 0 σ∗^2 = ∫∞ −∞ ( x−μ σ ) ...

The cumulative distribution function F(x)associated with the normal probability density function f(x) = 1 σ √ 2 π e− (^12) (x−μ) ...

Standardization The normal probability density function, sometimes called the Gaussian distri- bution , has the form N(x;μ, σ^2 ...

Figure 11-9. Standard normal probability curve and distribution function as area. Note that with a change of variable F(x) = Φ ( ...

The normal distribution function Φ(z) = √^1 2 π ∫z −∞ e−ξ (^2) / 2 dξ and the error function^1 which is defined erf(z) = √^2 π ∫ ...

shape of the original distribution being sampled. The normal distribution is also related to least-square estimation. It is also ...

represent the binomial coefficients in the binomial expansion (p+q)n= ( n 0 ) pn+ ( n 1 ) pn−^1 q+ ( n 2 ) pn−^2 q^2 +···+ ( n x ...