(a) Given that X- Bin(n. p) . Show that (i) the mean of X. is np (ii) the variance of X is np(1- p)

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Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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4.
The probability mass function, , of a discrete random variable, X, is defined by
kr, x= 1,2,3, 4,5
S(x) =-
*(10-x), x 6,7,8,9
Calculate
(i)
The value of the constant k
(ii)
The mean and variance of X
(ii)
E(3.X +3) and l'ar (4 X + 7)
5.
A continuous random variable V. has the probability density function f defined by
S(x) = k(3+ 2x-x*), 0sxs3
elsewhere
0,
Calculate
(i)
the value of the constant k.
(ii)
the mean and variance of N.
(iii)
the mode of X.
Given that the lower quartile is m, show that n satisfies the equation 4m' -12m -36m + 27 = 0
6.
(a)
Given that X - Bin(n. p). Show that
(i)
the mean of X, is np
(ii)
the variance of X is np(1- p)
(b) , It is known that 2 per thousand of identification cards (IDs) issued by a certain police oflice contains errors. In a certain
month, 1000 cards were produced. Find the probability to 4 decimal places, that
(i)
exactly 2 IDs will contain errors.
(ii)
at least 2 IDs will contain errors.
In this particular month, what is the most likely number of IDs that will contain errors?
Transcribed Image Text:4. The probability mass function, , of a discrete random variable, X, is defined by kr, x= 1,2,3, 4,5 S(x) =- *(10-x), x 6,7,8,9 Calculate (i) The value of the constant k (ii) The mean and variance of X (ii) E(3.X +3) and l'ar (4 X + 7) 5. A continuous random variable V. has the probability density function f defined by S(x) = k(3+ 2x-x*), 0sxs3 elsewhere 0, Calculate (i) the value of the constant k. (ii) the mean and variance of N. (iii) the mode of X. Given that the lower quartile is m, show that n satisfies the equation 4m' -12m -36m + 27 = 0 6. (a) Given that X - Bin(n. p). Show that (i) the mean of X, is np (ii) the variance of X is np(1- p) (b) , It is known that 2 per thousand of identification cards (IDs) issued by a certain police oflice contains errors. In a certain month, 1000 cards were produced. Find the probability to 4 decimal places, that (i) exactly 2 IDs will contain errors. (ii) at least 2 IDs will contain errors. In this particular month, what is the most likely number of IDs that will contain errors?
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