Random Variable and Approximately Gamma Distribution

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Assignment#3

Deadline:
For DL Students: 15th march
For Regular Students: 10th march

Source: Textbook
Q 4-5,
Find xu for u= 0.1, 0.2 … 0.9
a) if x is uniform in the interval (0,1);
b) if f(x)= 2e-2x U(x)

Q 4-7,
Show that if the uniform variable x has an Erlang density with n=2, then Fx(x) = (1-e-cx-cxe-cx) U(x)

Q 4-8,
The random variable x is N (10; 1), Find f (x | (x-10)2 <4)

Q 4-9,
Find f(x) if F(x) = (1-e-ax) U(x-c).

Q 4-10,
If x is N (0, 2) find
a) P{1≤ x ≤ 2}
b) P{1≤ x ≤2 | x ≥ 1}

Q4-14,
A fair coin is tossed 900 times and the random variable x equals the total number of heads. a) Find fx(x), 1: exactly, 2: approximately Gamma Distribution eq. b) Find P {435 ≤ x ≤ 460}.

Q4-25,
If P (A) = 0.6 and k is the number of successes of A in n trials, a) Show that P {550 ≤ k ≤ 650} = 0.999, for n=1000.
b) Find n such that P {0.59n ≤ k ≤ 0.61n} = 0.95

Q 4-26,
A system has 100 components. The probability that a specific component will fail in the interval (a, b) equals e-a/T – e-b/T. Find the probability that in the interval (0, T/4), no more than 100 components will fail.
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