Explain why not valid PDF

Questions that ask to identify and explain why a given function or graph cannot be a valid probability density function (violates non-negativity or total area condition).

6 questions

CAIE S2 2011 June Q4
4

  1. \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_611_712_1466_358}
    \includegraphics[max width=\textwidth, alt={}, center]{7c9a87ac-69c6-4850-82aa-8235bba581e8-2_618_716_1464_1155} The diagrams show the graphs of two functions, \(g\) and \(h\). For each of the functions \(g\) and \(h\), give a reason why it cannot be a probability density function.
  2. The distance, in kilometres, travelled in a given time by a cyclist is represented by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { 30 } { x ^ { 2 } } & 10 \leqslant x \leqslant 15
    0 & \text { otherwise } \end{cases}$$
    1. Show that \(\mathrm { E } ( X ) = 30 \ln 1.5\).
    2. Find the median of \(X\). Find also the probability that \(X\) lies between the median and the mean.
CAIE S2 2017 March Q5
5

  1. \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_292_517_264_338}
    \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_289_518_264_858}
    \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_273_510_365_1377} The diagram shows the graphs of three functions, \(f _ { 1 } , f _ { 2 }\) and \(f _ { 3 }\). The function \(f _ { 1 }\) is a probability density function.
    1. State the value of \(k\).
    2. For each of the functions \(\mathrm { f } _ { 2 }\) and \(\mathrm { f } _ { 3 }\), state why it cannot be a probability density function.
  2. The probability density function g is defined by $$g ( x ) = \begin{cases} 6 \left( a ^ { 2 } - x ^ { 2 } \right) & - a \leqslant x \leqslant a
    0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
    1. Show that \(a = \frac { 1 } { 2 }\).
    2. State the value of \(\mathrm { E } ( X )\).
    3. Find \(\operatorname { Var } ( X )\).
CAIE S2 2016 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534}
\includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ (a) Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    (b) Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    (c) Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_504_260_534}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_373_495_260_1123}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_497_776_534}
\includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.
  1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ (a) Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
    (b) Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
    (c) Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).
CAIE S2 2016 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431}
\includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.
  1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
  2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3
    0 & \text { otherwise } \end{cases}$$ (a) Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
    (b) Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
    (c) Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).
Edexcel S2 2018 October Q3
3. The function \(\mathrm { f } ( x )\) is defined as $$f ( x ) = \begin{cases} \frac { 1 } { 9 } ( x + 5 ) ( 3 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ Albert believes that \(\mathrm { f } ( x )\) is a valid probability density function.
  1. Sketch \(\mathrm { f } ( x )\) and comment on Albert's belief. The continuous random variable \(Y\) has probability density function given by $$g ( y ) = \begin{cases} k y \left( 12 - y ^ { 2 } \right) & 1 \leqslant y \leqslant 3
    0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
  2. Use calculus to find the mode of \(Y\)
  3. Use algebraic integration to find the value of \(k\)
  4. Find the median of \(Y\) giving your answer to 3 significant figures.
  5. Describe the skewness of the distribution of \(Y\) giving a reason for your answer.