Question

The random variable X follows a Poisson process with the given mean. Assuming

mu equals 6 commaμ=6,

compute the following.

​(a)​ P(66​)

​(b)​ P(Xless than<66​)

​(c) ​P(Xgreater than or equals≥66​)

​(d) ​P(55less than or equals≤Xless than or equals≤77​)

Answer 1

SOLUTION:

From given data,

The random variable X follows a Poisson process with the given mean. Assuming mu equals 6 comma compute the following. Please show work.

(a) P(6)

(b) P(X less than 6)

(c) P(X greater than or equals 6)

(d) P(5 less than or equals X less than or equals 7)

X Poisson ( )

P(X) = e^- * ^x / x !

= 6

P(X) = e^-6 * 6^x / x !

(a) P(6)

x = 6

P(6) = e^-6 * 6⁶ / 6 !

P(6) = 0.0024787* 46656/ 720

P(6) = 115.6486615 / 720

P(6) = 0.160623

(b) P(X less than 6)

P(X < 6) = P(0)+P(1)+P(2)+P(3)+P(4)+P(5)

P(X < 6) = e^-6 * 6⁰ / 0 !+e^-6 * 6¹ /1 !+e^-6 * 6² / 2 !+e^-6 * 6³ / 3 !+e^-6 * 6⁴ / 4 !+e^-6 * 6⁵ / 5 !

P(X < 6) = e^-6+ e^-6 * 6 /1 +e^-6 * 36 / 2+e^-6 * 216 / 6+e^-6 * 1296 / 24+e^-6 * 7776/ 120

P(X < 6) = 0.0024787+ 0.0148722+0.0446166+0.0892332+0.1338498+0.16061976

P(X < 6) = 0.445670

(c) P(X greater than or equals 6)

P(X > 6 ) = 1 - P(X < 6)

P(X > 6 ) = 1 - 0.445670

P(X > 6 ) = 0.55433

(d) P(5 less than or equals X less than or equals 7)

P(5 < X < 7) = P(5)+P(6)+P(7)

P(5 < X < 7) = e^-6 * 6⁵ / 5 !+e^-6 * 6⁶ / 6 !+e^-6 * 6⁷ / 7!

P(5 < X < 7) = e^-6 * 7776/ 120+e^-6 * 46656 / 720+e^-6 * 279936 / 5040

P(5 < X < 7) = 0.16061976+0.16061976+0.13767408

P(5 < X < 7) = 0.458913