Poisson Distribution and Random Variables

[doublecherrypie]

1.) Suppose a random variable X is uniformly distributed over [1,6]. The expected value of 1/X is... (the answer is ln(6) / 5)

Since E(x) = int ( x • f(x) ), I keep getting 1 underneath the integral sign. What am i doing wrong, please?

2.) A Geiger counter clicks, on the average, every 20 seconds. The number of clicks is a Poisson process. It is known that during one particular minute, the counter clicked at most 3 times. WHat is the probability that the number of clicks during that minute was at least 2? (the answer is 9 / 13)

3.) The number of clicks of a Geiger counter is a Poisson process. On teh average there is one click per minute. The geiger counter is turned on for one minute. Given there are at most three clicks, what is the expected number of clicks? (the answer is 15 / 16).

 

[royhaas]

1) Compute \(\displaystyle \int_1^6 (1/x)(dx/5)\).

2) \(\displaystyle P(X \>= 2| X \le 3)\)

3)\(\displaystyle E(X|X \le 3)\)

 

[doublecherrypie]

Could you possibly elaborate on how to do #3? Thank you SO MUCH

 

[marcmtlca]

\(\displaystyle E[X|X \geq 3]=\displaystyle\sum_{x=0}^{\infty}x \cdot P\left[X=x | X \geq 3\right]=\displaystyle\sum_{x=0}^{\infty}x \cdot \frac{P\left[(X=x) \cap (X \geq 3)\right]}{P(X \geq 3)}\)

 

[tkhunny]

at most three clicks

Your entire conditional distribution is defined by Pr(0), Pr(1), Pr(2), and Pr(3). Scale those to a complete distribution and calculate the expdctation.

If Scale = Pr(0)+Pr(1)+Pr(2)+Pr(3), then

E[X] = 0*Pr(0)/Scale + 1*Pr(1)/Scale + 2*Pr(2)/Scale + 3*Pr(3)/Scale

You just have to think it through. You cannot memorize a formula for everthing.