Find parameters from given statistics

4
\includegraphics[max width=\textwidth, alt={}, center]{65b50bfb-5fd8-4cf3-ae3b-cffc12e23cd8-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

4
\includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

4
\includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

4
\includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)

1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
2. find the value of \(\alpha\) and the value of \(\beta\)
3. find \(\operatorname { Var } ( X )\)
4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) where \(0 < a < b\)

Given that \(\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }\)

1. 1. show that \(\mathrm { E } ( X ) = \frac { 5 a } { 2 }\)
   2. find \(\mathrm { P } ( X > b - 4 a )\) The continuous random variable \(Y\) is uniformly distributed over the interval [3, c] where \(c > 3\) Given that \(\operatorname { Var } ( Y ) = 3 c - 9\), find
2. 1. the value of \(c\)
   2. \(\mathrm { P } ( 2 Y - 7 < 20 - Y )\)
   3. \(\mathrm { E } \left( Y ^ { 2 } \right)\)

1. The continuous random variable \(T\) is uniformly distributed on the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\)

Given that \(\mathrm { E } ( T ) = 2\) and \(\operatorname { Var } ( T ) = \frac { 16 } { 3 }\), find

1. the value of \(\alpha\) and the value of \(\beta\),
2. \(\mathrm { P } ( T < 3.4 )\)

1. Akia selects at random a value from the continuous random variable \(W\), which is uniformly distributed over the interval \([ a , b ]\)

The probability that Akia selects a value greater than 17 is \(\frac { 1 } { 5 }\) The probability that Akia selects a value less than \(k\) is \(\frac { 53 } { 60 }\)

1. Find the probability that Akia selects a value between 17 and \(k\) It is known that \(\operatorname { Var } ( W ) = 75\)
2. 1. Find the value of \(a\) and the value of \(b\)
   2. Find the value of \(k\)
3. Find \(\mathrm { P } ( - 5 < W < 5 )\)
4. Find \(\mathrm { E } \left( W ^ { 2 } \right)\)

1. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)

Given that \(\mathrm { P } ( 5 < X < 13 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 9\), find \(\mathrm { P } ( 3 X > a + b )\)
(ii) The continuous random variable \(Y\) is uniformly distributed over the interval \([ 1 , c ]\) Given that \(\operatorname { Var } ( Y ) = 0.48\), find the exact value of \(\mathrm { E } \left( Y ^ { 2 } \right)\)
(iii) A wire of length 20 cm is cut into 2 pieces at a random point. The longest piece of wire is then cut into 2 pieces, equal in length, giving 3 pieces of wire altogether. Find the probability that the length of the shortest piece of wire is less than 6 cm .

5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).

1. Write down the probability density function of \(X\), for all \(x\).
2. Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\). A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the standard deviation of \(X\),
5. the probability that the shorter piece of wire is at most 30 cm long.

3. The random variable \(X\) has a continuous uniform distribution on \([ a , b ]\) where \(a\) and \(b\) are positive numbers. Given that \(\mathrm { E } ( X ) = 23\) and \(\operatorname { Var } ( X ) = 75\)

1. find the value of \(a\) and the value of \(b\). Given that \(\mathrm { P } ( X > c ) = 0.32\)
2. find \(\mathrm { P } ( 23 < X < c )\).

4. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) Given that \(\mathrm { E } ( X ) = 3.5\) and \(\mathrm { P } ( X > 5 ) = \frac { 2 } { 5 }\)

1. find the value of \(\alpha\) and the value of \(\beta\) Given that \(\mathrm { P } ( X < c ) = \frac { 2 } { 3 }\)
2. 1. find the value of \(c\)
   2. find \(\mathrm { P } ( c < X < 9 )\) A rectangle has a perimeter of 200 cm . The length, \(S \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 30 cm and 80 cm .
3. Find the probability that the length of the shorter side of the rectangle is less than 45 cm .

1. David aims to catch the train to work each morning. The scheduled departure time of the train is 0830

The number of minutes after 0830 that the train departs may be modelled by the random variable \(X\). Given that \(X\) has a continuous uniform distribution over \([ \alpha , \beta ]\) and that \(\mathrm { E } ( X ) = 4\) and \(\operatorname { Var } ( X ) = 12\)

1. find the value of \(\alpha\) and the value of \(\beta\). Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
2. Find the probability that David will be late for work. Given that David is late for work,
3. find the probability that he overslept.

2 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b
0 & \text { otherwise } \end{cases}$$

1. Write down, in terms of \(a\) and \(b\), the value of \(k\).
2. 1. Given that \(\mathrm { E } ( X ) = 1\) and \(\operatorname { Var } ( X ) = 3\), find the values of \(a\) and \(b\).
   2. Four independent values of \(X\) are taken. Find the probability that exactly one of these four values is negative.
      [0pt] [3 marks]

2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.

1. Show that \(n = 23\) and find the variance of \(X\).
2. Find \(\mathrm { P } ( 10 < X < 14 )\).

2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).

1. It is given that \(W\) has a Poisson distribution.
   (a) Write down the standard deviation of \(W\).
   (b) Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
2. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).

4 A random variable \(X\) has a rectangular distribution. The mean of \(X\) is 3 and the variance of \(X\) is 3
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1. Determine the probability density function of \(X\).
   Fully justify your answer. 4
2. A 6 metre clothes line is connected between the point \(P\) on one building and the point \(Q\) on a second building. Roy is concerned the clothes line may break. He uses the random variable \(X\) to model the distance in metres from \(P\) where the clothes line breaks. 4
3. 1. State a criticism of Roy's model. 4
4. (ii) On the axes below, sketch the probability density function for an alternative model for the clothes line.
   \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-05_584_1162_1210_438}