Determine k, if the p.d.f of the random variable is given by f(x) = {ke^-2x , 0 ≤ x ≤ infinity,
0 otherwise Find P(0 < x < log√5 )
Answers
Rather than summing probabilities related to discrete random variables, here for
continuous random variables, the density curve is integrated to determine probability.
Exercise 3.1 (Introduction)
Patient’s number of visits, X, and duration of visit, Y .
0 1 2 3
0.25
0.75
1
4
probability =
value of function,
F(3) = P(Y < 3) = 5/12
0 1 2 x
0.25
0.50
0.75
1
0 1 2
0.25
0.50
0.75
1
density, pmf f(x)
probability (distribution): cdf F(x)
probability less than 1.5 =
sum of probability
at specific values
P(X < 1.5) = P(X = 0) + P(X = 1)
= 0.25 + 0.50 = 0.75
P(X = 2) = 0.25
0 1 2 3
1/3
1/2
2/3
1
4
density, pdf f(y) = y/6, 2 < y < 4
probability less than 3
= area under curve,
P(Y < 3) = 5/12
x
probability at 3,
P(Y = 3) = 0
probability less than 1.5 =
value of function
F(1.5 ) = P(X < 1.5) = 0.75
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Figure 3.1: Comparing discrete and continuous distributions