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Chapter 3: Q13E (page 107)

An ice cream seller takes 20 gallons of ice cream in her truck each day. LetXstand for the number of gallons that she sells. The probability is 0.1 thatX=20. If she doesn’t sell all 20 gallons, the distribution ofXfollows a continuous distribution with a p.d.f. of the form


wherecis a constant that makes Pr(X <20)=0.9. Find the constantcso that Pr(X <20)=0.9 as described above.

Short Answer

Expert verified

The computed value of c is 0.0045.

Step by step solution

01

Given the information 

X is the random variable for the number of gallons sold by the seller of 20 gallons ice cream.

The probability that 20 gallons is sold is 0.1.

The random variable X follows a continuous distribution with a probability density function is,


The probability is given as P(X< 20)=0.9

02

Calculate the value of c

From the provided data, the value of c is computed as,

Therefore, the value of c is 0.0045.

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Most popular questions from this chapter

A civil engineer is studying a left-turn lane that is long enough to hold seven cars. LetXbe the number of cars in the lane at the end of a randomly chosen red light. The engineer believes that the probability thatX=xis proportional to(x+1)(8−x)forx=0, . . . ,7 (the possible values ofX).

a. Find the p.f. ofX.

b. Find the probability thatXwill be at least 5.

There are two boxes A and B, each containing red and green balls. Suppose that box A contains one red ball and two green balls and box B contains eight red balls and two green balls. Consider the following process: One ball is selected at random from box A, and one ball is selected at random from box B. The ball selected from box A is then placed in box B and the ball selected from box B is placed in box A. These operations are then repeated indefinitely. Show that the numbers of red balls in box A form a Markov chain with stationary transition probabilities, and construct the transition matrix of the Markov chain.

Question:LetYbe the rate (calls per hour) at which calls arrive at a switchboard. LetXbe the number of calls during at wo-hour period. A popular choice of joint p.f./p.d.f. for(X, Y )in this example would be one like

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{{{\left( {{\bf{2y}}} \right)}^{\bf{x}}}}}{{{\bf{x!}}}}{{\bf{e}}^{{\bf{ - 3y}}}}\;{\bf{if}}\;{\bf{y > 0}}\;{\bf{and}}\;{\bf{x = 0,1, \ldots }}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

a. Verify thatfis a joint p.f./p.d.f. Hint:First, sum overthexvalues using the well-known formula for thepower series expansion of\({{\bf{e}}^{{\bf{2y}}}}\).

b. Find Pr(X=0).

Suppose that a random variableXhas the uniform distributionon the integers 10, . . . ,20. Find the probability thatXis even.

Consider the situation described in Example 3.7.14. Suppose that \(\) \({X_1} = 5\) and\({X_2} = 7\)are observed.

a. Compute the conditional p.d.f. of \({X_3}\) given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\).

b. Find the conditional probability that \({X_3} > 3\)given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\)and compare it to the value of \(P\left( {{X_3} > 3} \right)\)found in Example 3.7.9. Can you suggest a reason why the conditional probability should be higher than the marginal probability?
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