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1) (4 mark) Let X be a Poisson random variable with a certain parameter A > 0 such that P(X = 2) = P(X = 3)

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1) (4 mark) Let X be a Poisson random variable with a certain parameter A > 0 such that P(X = 2) = P(X = 3). Find P(X > 2). Hint: use the complement rule. Show all the steps. Circle the final answer. 

2) (3 marks) A continuous random variable X has the following probability density function f„(x) = k e-x2/4, —co < x < 00. Find the value of the constant k. Hint: compare with the density of normal distribution. Show all the steps. Circle the final answer. 
 

3) (4 marks) A continuous random variable X has the following probability density function 
f e-s}-2 for x > 2 fx (x) 0 for x< 2 
Find the median of the distribution of X. That is, find m for which P(X < m) = 2. Show all the steps. Circle the final answer. 
 

4) (4 marks: 1, 3) The following table summarizes the joint probability mass function of the discrete random variables X and Y: 
—2 3 5 0 1/15 2/15 1/15 y 2 3/15 2/15 1/15 4 2/15 1/15 2/15 Show all the steps. Circle the final answers. 1. Find the joint cumulative distribution function F„,,, (4, 1.8). 
2. Find the conditional probability P(Y = 21Y < 3). 

5) (5 marks) Let X 

If Y = In (X + 1); Y. Don't forget to the steps. Circle 
1°) Al Read aloud V Draw 

`9 Highlight \/ 
be a continuous random variable with the following probability density function f2 ± X > 0 0' otherwise 
use the "distribution function technique" to find the probability density function of specify where the probability density function of Y is greater than zero. Show all the final answer.