# Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f. f (x) = k (4–x2), for –2 ≤ x ≤ 2 and = 0 otherwise. P(x > 0) - Mathematics and Statistics

Sum

Suppose error involved in making a certain measurement is continuous r.v. X with p.d.f.

f (x) = k (4 – x^2 ), for –2 ≤ x ≤ 2 and = 0 otherwise.

P(x > 0)

#### Solution

Since, f is the p.d.f. of X,

 int_(-∞)^∞ f (x) dx = 1

∴  int_(-∞)^-2 f (x) dx + int_(-2)^2 f (x) dx +  int_(2)^∞f (x) dx= 1

∴ 0 +  int_(-2)^2 k (4 -x^2) dx = 1

∴ k  int_(-2)^2  (4 -x^2) dx = 1

∴ k [ 4x - x^3/3]_-2^2 = 1

∴ k [(8-8/3)-(-8+8/3)]= 1

∴ k(16/3+16/3) = 1

∴ k(32/3) = 1

∴ k = 3/32

P(x > 0)

=  int_(0)^∞ f (x) dx

=  int_(0)^2 f (x) dx+  int_(2)^∞ f (x) dx

=  int_(0)^2 k (4-x^2) dx+ 0

= k int_(0)^2  (4-x^2) dx

=3/32[4x -x^3/3]_0^2  ..........[∵ k=3/32]

=3/32 [8-8/3] = 3/32 xx16/3 = 1/2

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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 7 Probability Distributions
Exercise 7.2 | Q 7.1 | Page 239
Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 7 Probability Distributions
Miscellaneous Exercise | Q 13.1 | Page 244