(3) Find the density of the random variable X representing the number of correctly placed letters when n = 4 letters are randomly placed into the four envelopes The following answers are proposed for the density, Values of X 1 2 3 4 Probabilities Po Pi P2 P3 P4 where a po = PI = P2 = % P3 = 0,p4 = , tb) po = P1 = P2 = p3 = 0, p4 = . c) po = }.P1 = P2 =, P3 = , P4 = . Pi = P2 = P3 = P4 = %3D %3D %3D (d) Po = (e) None of the above Then correct answer is: (a) (b) (c) (d) (е) N/A (Select One)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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Number of letters = 4
Total number of ways in which 4 letters can be placed in 4 envelopes = 4! = 24
X denotes the number of correctly placed envelopes.
Then,
Number of ways in which no letter is placed in the correct envelope (X=0) is:
Therefore,
Number of ways in which only 1 letter is placed in the correct envelope (X=1) is:
Therefore,
Number of ways in which 2 letters are placed in the correct envelope (X=2) is:
Therefore,
Number of ways in which 3 letters are placed in the correct envelope (X=3) is: 0 as out of 4 total envelopes, there can not be only 1 wrong letter in the wrong envelope
Therefore,
Number of ways in which all 4 letters are placed in the correct envelope (X=4) is: 1
Therefore,
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